Regular Graph Synthesis, Visualization & Recognition.

Roger Beresford. 2004, revised 27th. November 2007.

"The theory of symmetric cubic graphs is full of strange delights." Norman Biggs.

1. Summary.

Regular graphs have the same number edge of edges meeting at each vertex (edges join different vertices). In cubic graphs edge = 3. This notebook investigates the creation, identification and display of small regular graphs with edge = 3, 4, 5, 6, 7 or 8, in the Combinatorica graph format. A procedure allows the development of regular graphs as several circuits (which may be stellated) with interconnecting edges, or by a generalization of the LCF (Lederburg-Coxeter-Frucht) [1] procedure, or as generalized Petersen-type graphs. Edge deletion and insertion routines allow the creation of modifiied graphs. Hamiltonian representations (where they exist, with all the vertices on a closed circuit) and near-Hamiltonian representations (a closed circuit plus a few inner vertices) can be found; these provide different "embeddings" (isomorphic visualizations) of the graphs. Many graphs have elegant structured visualizations, often symmetrical. Hamiltonian forms often lack visual appeal.
	Over 810 regular embeddings (over 630 different graphs) are supplied in graphnData, (n=edge, the number of edges at each vertex). They are ordered by the number of vertices and then by the number of automorphisms (symmetries). Some are random examples; many are from the cited references. Isomorphs are listed together. 72 graphs with recognised names are included. regGraphId[g] identifies any isospectral embedding of graphs in the database, by comparisons of factorized characteristic polynomials. (Isospectral non-isomorphic regular graphs are unlikely but two known (K=4,v=10) pairs, and some others, are indicated by names ending in "?".)  graphNo[ edge, size, index, isomorph] creates the corresponding embedding for tests or display. 
	A few disconnected graphs and "peninsula graphs" (regular non-hamiltonian graphs with bridge edges) are included.
	Procedures are supplied to test the correctness of the data, and to show a "Gallery of Graphs". 
	An experimental procedure allGraphs attempts to create all graphs with a specified form. It has been used to find and store a number of graphs. The large numbers of regular graphs with more than 14 vertices means that this procedure has to be severely constrained on a PC.
	A second experimental procedure subGraph (inspired by [5]) attempts to remove edge sets from a known graph, and add others to restore regularity.
	PPn[n] creates the projective plane of order n (if n is prime), and represents it as a 2n+2-regular graph with coloured lines (sets of edges such that each edge occurs in exactly one line). Data for PP4, PP8, & PP9 (modified from Moorhouse [6]) is included and demonstrated.
	Please inform me of any errors, and provide details of any graphs or embeddings that you wish to have added to the data, at rhberesford@btinternet.com.

2. Version, Revisions, Notes.

2.1 Version.

A preliminary version was dated 24/2/4. Version 1, using Mathematica 5 (Mathematica 4.2 OK apart from EWIsomorphicQ), first dated 4/5/4. Cuboctahedral, Clebsch, etc & multiple additions & deletions added 5-11/4/4. Peninsula graphs added 12/4/4. Graph names rationalized, more graphs included 14/4/4. EWIsomorphic corrected, many graphs found to be isomorphic, 16/4/4. newgraphs used to fill some gaps, 18/4/4. Bridge graphs (example 16) corrected; newgraphs handles 4,5,6-regular graphs. Many added. 19/4/4. Characteristic Polynomial, regGraphId, regGraphOptions 23/4/4. allGraphs added 2/5/4, replacing newgraphs. Used to give many new graphs 4/5/4. gS addEdge correction, additional graphs 4/10/4.
Many 7-regular graphs, including Hoffman-Singleton [5], added; subGraph created in the 19/11/4 revision.
26/11/4 More HS-subgraph symmetrical plots found.
27/11/2007 More graphs added, including some 8-regular graphs and hyper-symmetric graphs.
Projective Planes added, with a procedure to create them for prime n and to plot them ascoloured lines (sets of n edges) in 2n-regular graphs.

2.2 Keywords.

Regular graphs, Hamiltonian cycles, LCF, stellations, generalized Petersen graphs, graph embeddings, peninsula graphs, Projective Planes.

2.3 Warnings.

"Global" effects - comb1, comb2, crash, Hc, incant, mx, tf, tst, tst1, vertx, x are altered by 
various routines. ShowLabelledGraph and vertx use different labellings. Only projective planes with n a prime can be created, though PP4, PP8, & PP9 tables are supplied.

2.4 Limitations.

The lists of graphs are far from complete. The number of automorphisms is shown as 0 in the data for some graphs, as it has not been calculated. The incant list is wrong for multi-circuit graphs. PPn vertex colouring needs improving.

3. Acknowledgements, References.

This work was inspired by Ed Pegg's description of symmetric cubic graphs in Mathgames [2], but the procedures have the flexibility to create many others.
	An improved version of MyIsomorphicQ is based on Eric Weisstein's Mathworld graphs package [3] (which could be wrong because Eccentricity gives an unsorted list). [3] also provided some LCF notations. hamiltonianCycle is modified from Skiena [4] p197. More interesting regular Mathworld graphs added 2007.

[1] http://mathworld.wolfram.com/LCFNotation.html

[2] Ed Pegg Jr, www.maa.org/editorial/mathgames/mathgames_12_29_03.html

[3] http://mathworld.wolfram.com/graphs

[4] Steven Skiena, Implementing Discrete Mathematics, Addison Wesley, 1990.

[5] Ed Pegg Jr, www.maa.org/editorial/mathgames/Hoffman.gif

[6] http://www.uwyo.edu/moorhouse/pub/planes/pg24.txt etc

4. Usage.

allGraphs::Usage="allGraphs[ed_,h_, n1_:0, n0_:{{},1}, longest_:10, shortest_:2] attempts to create all graphs with valency=ed, a Hamiltonian cycle of h vertices, an optional inner cycle of n1 vertices, and the crosslinks specified by n0. The length of arcs can be restricted by setting longest and shortest, to reduce the search space. Permitted edges are found, and added recursively until the graph is complete or impossible; Unknown graphs are stored, together with thir incantations and characteristic polynomials.";

addEdge::Usage="addEdge[j,k] is called by other procedures to insert the specified edge into the comb1 & vertx lists. It returns True if successful, and False if one of the vertices already has a full complement of edges; this halts the current edge creation loop.";

colo::Usage="A selection of named colours used by PPn. Very light and very dark colours are omitted";

cp::Usage="cp[graph] calculates the characteristic polynomial for the specified regular graph; the (1-edge) term is cancelled out to reduce storage requirements.";

delEdge::Usage="delEdge[Negated vertex index x, vertex index y] removes the specified edge from x to y. If the optional parameters are supplied, the edges x+i dx, y+i dy are also deleted, i=={i,.., n-1}.";
 
EWIsomorphicQ::Usage = "A copy of MyIsomorphicQ from Eric Weisstein's Mathworld Graph package, with Eccentric corrected";

gallery::Usage="gallery[GraphList,startindex,breadth,depth]" creates an array of graphs once a list has been created (as in section 7). The lists G3 to G8 give all the graphs in the database (Projective Planes are omitted for n>4). The first element of each data entry is used as the PlotLabel; this is the name if the graph has one, or the valency followed by the number of edges. Then comes the number of automorphisms, and an index if necessary. Blue graphs indicate a change in the number of vertices; green graphs indicate a new graph, and black graphs are different embeddings of the previous coloured graph.";

genPetersen::Usage="genPetersen[orbit o, stellation s] creates a generalization of the Petersen graph as a cubic graph. If both parameters {o,s} are positive, the graph is two linked orbits, with the inner orbit stellated. If they are equal and negated, 3 orbits are created with lengths {-o,-2o,-o}. Other cases are shown in Tutorial example 14. ";

gP::Usage="Abbreviation for genPetersen";

graphNo::Usage = "graphNo[ed, no of vertices, index number, isomorph number] creates the specified graph, using the LCF procedure if the graphNData is a list and power (a common form), otherwise it uses ToExpression[string Data]."; 

graphSynthesis::Usage="graphSynthesised[ed_,c_,croslnk_] creates an undirected graph in Combinatorica format. The graph will have (at most) ed edges at each vertex (vertices with fewer edges will be reported if pr>2). It will have one or more circuit with c[[i,1]] radius, |c[[i,2]]| points, c[[i,3]] stellation step, c[[i,4]] offset angle, |c[[i,5]]| link vertex offset. The circuit is unlinked if the stellation step is zero, or linked by edges of the specified step length (the default value 1 gives a closed circuit). Unless c[[i,5]] is negative, each circuit is automatically linked (symmetrically as far as possible) to the previous circuit, or (if c[[i,2]] is negative) to the one before that.
The croslnk data defines other edges, and can be in (1) the LCF format, as a list of edge lengths for the edges leaving vertices 1,2,... followed by a power (the number of times this list is repeated); (2) a list of 3-element-lists {first vertex number, edge step length, offset from the end of the edge to the start of the next, in the same circuit}, with each list being implemented until the next edge would involve a full vertex; (3) 2 or 5 vertex indices, the first being positive, indicating an edge to be added, with 3rd and 4th parameters being increments that are applied to the edge numbers for the number of times specified by the 5th.parameter (if the second is negative, its absolute value is used, and any number of edges can meet at a vertex); or (4) 2 or 5 vertex indices, the first being negative, indicating an edge to be deleted, with 3rd and 4th parameters being increments that are similarly applied to delete the number of edges specified by the 5th.  Types 2,3 & 4 can be intermingled. Types 3 & 4 do not (yet) increment within the same circuit.
The graph is built up as three lists:- vertx is the indices of the vertices linked to vertex[i] (used by hamiltonianCycle & replot); comb1 & comb2 are lists of edges and points that are finally combined to give the Combinatorica  'Graph' structure.";

gS::Usage="Abbreviation for graphSynthesis";

gShort::Usage="converts a PPn graph to an un-coloured form that can be tested by regGraphId etc.";

hamiltonianCycle::Usage="hamiltonianCycle[n] uses the data in vertx to find different circuits starting at vertex 1. If there are none, 'Not connected' is returned. If none are Hamiltonian (i.e. with the last vertex linking to the first vertex) the number of non-Hamiltonian cycles is reported, and they are supplied in the global variable Hc. If the graph is Hamiltonian, a maximium of n such circuits are sought. They are supplied in Hc. ";

infill::Usage="allGraphs uses infill to recursively add chords";

plTopp::Usage="Converts a line list to a pp array";

PPn::Usage="PPn[order,centre_:{{0,0}}] creates (if order is prime) the correspondingProjective Plane as an adjacency matrix pp, a line list pl, and a 2n-regular graph. The graph consists of a point at centre and rays of points joined by edges, coloured to show line membership. Each possible edge appears in exactly one line. If pr is negative, a table ppTest (examples pp4, pp8, pp9 supplied) is imported after the graph framework has been created";

regGraphId::Usage = "regGraphId[graph,c_:False]; Combinatorica graphs need c=True to create the specified regular graph and generate vertx. vertx is used to test for regularity before testing for isomorphism against the known graphs of that valency and size. If a match is found, the valency, size, & index are returned, together with the graph name (either as {valency, size, index, no. of Automorphisms} or as {Name, no. of Automorphisms}). EWIsomorphicQ was originally used, but was inefficient as it always needed to recalculate the CharacteristicPolynomial for each comparison; this is now stored.";

replot::Usage="replot[n,a] creates a graph from the n'th circuit in Hc, as produced by hamiltonianCycle. If this is not Hamiltonian, vertices are moved into the centre until a closed outer loop is found. Central vertices are placed on a circle, which can be rotated by an angle a to improve the appearance.";

subGraph::Usage="subGraph[vertexlist,depth,inlist,graph:tst] expands the vertexlist to include neighbours to the specified depth, and selects (positive depth) or deletes (non-positive depth) them before adding the inlist edges. The graph is stored as tst1 and is plotted unless depth is real.";testGraph::Usage="testGraph is used by allGraphs & infill to test for unknown graphs and to store new ones in results & incant. incant is still wrong for multi-circuits.\!\(\*
StyleBox["\"",
FontSlant->"Plain"]\);Testpp::Usage="Checks that each possible edge occurs in exactly one line in pp.";

5. Programmes and Data, Version dated 27/11/7. (Initialize this section.)

Preliminaries.

Off[General::"spell", General::"spell1", 
      Part::"partd"]  
tf = TraditionalForm;
<< DiscreteMath`Combinatorica`;

namedGraphs = {Petersen, Franklin, Bidiakis - cube, A4, Frucht, 
        Heawood, MoebiusKantor, Pappus, Desargues, Dodecahedron, FlowerSnark, 
        F24, TruncatedOctahedron, McGee, TruncatedCube, F26, Coxeter, Levi, 
        Dyck, F38, F40, FlowerNonSnark, F42, F48, GreatRhombicuboctahedron, 
        F54, Gray, F56a, F60, TruncatedIcosahedron, F62, F64, Harries, 
        Balaban10, F72, F74, Goldberg, F90 Foster, Biggs - Smith, Balaban11, 
        Tutte, AntPrism3, AntPrism4, AntPrism5, Cuboctahedron, AntPrism6, 
        Chvatal, Antiprism7, C4C4, Antiprism8, Antiprism9, Robertson, 
        Folkman, OddO4, SmallRhombicuboctahedron, cage4_  6, Doyle, 
        Icosidodecahedron, SmallRhombicosidodecahedron, Meredith, 
        Icosahedron, SnubCube, Meringer, Wong, Foster 5 - 5 - cage, 
        Robertson - Wegner, Wells, Sylvester, 16 - cell, Klein, 
        Hoffman - Singleton, PP3}; Length[namedGraphs]

pr = 0;

Programmes.

Graph Synthesis.

Clear[graphSynthesis];graphSynthesis[ed_:  3, 
          c_: {{ .5, 5, 2}, {1. , 5}}, croslnk_: {}]:=Module[{k, l, m, m0, m1, n = 0, n0, ns = {}, n1, n2, n3, n4, n5, 
              p, ro, s, s1, s2, t0, t1, t2},
  comb1 = {};comb2 = {};vertx = {};mx = ed;crash = 0;

  
Do[If[Length[c[[i]]] < 2,Print["Faulty c data."];Abort[]];
n0 = n + 1;AppendTo[ns, n0];r = Abs[c[[i, 1]]];n1 = Floor[Abs[c[[i, 2]]]];ro = 
                      If[Length[c[[i]]] > 3 && NumberQ[c[[i, 4]]], 
                        c[[i, 4]], 0];
Do[n++; AppendTo[vertx, {}]; 
                      AppendTo[comb2, 
                        Chop[{{r Cos[2.  π j/n1 - ro], 
                              r Sin[2.  π  j/n1 - ro]}}, 
                          10^-5]], 
{j, 0, n1 - 1}];
s1 = 
                      If[Length[c[[i]]] > 2 && NumberQ[c[[i, 3]]], 
                        Abs[c[[i, 3]]], 1, 1];arc = 
                      If[IntegerQ[c[[i, 1]]] && s1 == 1, True, 
                        False];
  m0 = 
                      If[s1 > 1 && s1 == n1/2, n1/2 - 1, n1 - 1];
If[n1 < 3 || s1 == 0, , 
                      Do[If[addEdge[
                            p + n0, 
Mod[p + s1 + n0, n1, 
                              n0]], , Break], {p, 0, m0}]],
{i, Length[c]}];AppendTo[ns, n + 1];arc = False;
(*ns now contains the orbit starting points*)

   
Do[
                n5 = If[Length[c[[i]]] > 4, c[[i, 5]], 
                    0]; 
If[n5 ≥ 0, 
                  n1 = Abs[c[[i, 2]]]; 
                  n3 = If[Length[c[[i]]] > 2, c[[i, 3]], 1]; 
                  n4 = If[c[[i, 2]] < 0, -1, 0]; n0 = ns[[i]]; 
                  m1 = c[[i - 1 + n4, 2]]; 
                  m0 = ns[[i + n4]] - m1; 
Switch[n1, 
                    1, If[n3 ≠ 0, 
                      Do[addEdge[n0, 
                          Mod[Floor[j m1/ed + 1] + n5, m1, m0]], {j, 0, 
                          ed - 1}]], 
2, 
                    addEdge[n0, n0 + 1]; 
                    Do[addEdge[Floor[n0 + (j - 1)/(ed - 1)], 
                        Mod[Floor[j  m1/2/(ed - 1)] + n5, m1, 
                          m0]], {j, 2  ed - 2}], 
any_, 
                    Do[Do[
                        If[addEdge[n0 + p, 
                            Mod[Floor[p m1/n1] + j + 1 + n5, m1, m0]], , 
                          Break], 
{p, 0, 
                          n1 - 1}], 
{j, 0, 
                        ed - 2}]]], 
{i, Length[c], 
                  2, -1}];

  
If[Length[croslnk] ≥ 2 && 
                    NumberQ[
                      m0 = croslnk[[2]]],  
s1 = 
                      If[Length[croslnk] > 2 && 
                          NumberQ[
                            croslnk[[3]]], croslnk[[3]], 0, 0];k = 1;Do[Do[m = 
                                croslnk[[1, 
                                  j]];If[! IntegerQ[m], 
                                  Print["Faulty LCF data."]; Abort[]];If[
                                m ≠ 0, 
                                addEdge[Mod[k + s1, n, 1], 
                                  Mod[k + m + s1, n, 1]]];k++,{j, 
                              Length[
                                croslnk[[1]]]}],{i, m0}],
  
Do[If[! IntegerQ[(j = 
                                        croslnk[[i, 
                                        1]] ) (k = 
                                        croslnk[[i, 
                                        2]])], 
                              Print[{"Faulty non-integer croslnk data.", 
                                  j, k}]; Abort[]];

If[j < 0,  
If[
                                Length[ croslnk[[i]]] == 2, 
                                delEdge[-j, 
                                  k],   
Do[delEdge[-j, k]; j -=  croslnk[[i, 3]]; 
                                  k +=  croslnk[[i, 4]], { 
                                    croslnk[[i, 5]]}]],
  s = Sign[k];k = Abs[k];
If[Length[ 
                                        croslnk[[i]]] == 2,addEdge[j, k, s],
If[! IntegerQ[
                                        m = croslnk[[i, 
                                        3]]], 
                                        Print["Faulty m croslnk data"]; 
                                        Abort[]];

If[(Length[
                                        croslnk[[i]]] == 5) && 
                                        j > 0,  (*Stepped addition,  -k allows order increase*) 

Do[
                                        addEdge[j, k, s]; 
                                        j = Mod[j +  croslnk[[i, 3]], n, 
                                        1]; k = 
                                        Mod[k +  croslnk[[i, 4]], n, 
                                        1], { 
                                        croslnk[[i, 
                                        5]]}], 

h = 2; While[j ≥ (n1 = 
                                        ns[[h]]), h++]; m0 = ns[[h - 
                                        1]]; 
                                        m1 = n1 - m0; 
h = 
                                        2; While[
                                        croslnk[[i, 
                                        2]] + 
                                        croslnk[[i, 
                                        1]] ≥ (t1 = 
                                        ns[[
                                        h]]), h++]; 
                                        t0 = 
                                        ns[[h - 
                                        1]]; 
                                        t1 = 
                                        t1 - t0; 
While[
                                        h = Mod[j + k, t1, t0]; 
                                        addEdge[j, h], 
                                        j = Mod[h + m, m1, m0]]]]],
{i, Length[croslnk]}];
];

If[ed > 1,  miss = {};
Do[
                      If[Length[vertx[[i]]] ≠ ed, 
                        AppendTo[miss, i]], {i, n}];If[Length[miss] > 0,If[pr > 2, 
                            Print[{"Incomplete vertices", miss}], , ]];
tst = 
                      Graph[comb1, comb2, regGraphOptions]]];

gS[any__]:=graphSynthesis[any];

addEdge[j_, k_, s_:  1]  :=
Module[{},If[crash++ > 9999, Abort[]];If[
                j == k || MemberQ[vertx[[j]], k], , 
                If[s < 0 || (Length[vertx[[j]]] < mx && 
                        Length[vertx[[k]]] < 
                          mx), 
AppendTo[vertx[[j]], 
                    k]; AppendTo[vertx[[k]], j]; 
                  AppendTo[comb1, {{Min[j, k], Max[j, k]}}]; True, 
                  False]]];

delEdge[j_, k_]:=
Module[{},If[crash++ > 9999 , Print["delEdge crash"]; 
                  Abort[]];
vertx[[j]] = 
                Delete[vertx[[j]], Position[vertx[[j]], k]];vertx[[k]] = 
                Delete[vertx[[k]], Position[vertx[[k]], j]];comb1 = 
                Delete[comb1, 
                  Position[comb1, {{Min[j, k], Max[j, k]}}]]];

handFill allows a set of edges to be added to an incomplete current graph, to show it, and to identify it if the result is regular.

handFill[e_]:=Module[{}, 
          Do[addEdge[e[[i, 1]], e[[i, 2]]], {i, Length[e]}]; 
          ShowLabeledGraph[tst = Graph[comb1, comb2, regGraphOptions]]; 
          regGraphId[tst]];

Hamiltonian Cycles, replot, vertx.

hamiltonianCycle is modified from Skiena p197 HamiltonianCycle to use vertx by default and to provide more than one circuit or, in non-Hamiltonian cases, near-circuit. The specified number of circuits, or all the non-Hamiltonians, are stored in Hc. They can then be plotted via replot[n]. The message "Not connected?" is output if no circuits are found. This occurs with some awkward connected non-Hamiltonian graphs, as well as with unconnected graphs.

hamiltonianCycle[count_Integer:  1, 
        vert_:  vertx]:=Module[{s = {1}, all = {}, c = count, done, x, v, ve = vert, ind, 
            n = Length[vert], NHc = {}},If[! MatrixQ[ve], Print["Not a vertex list"]; 
                Abort[]];
ind = Table[1, {n}];Hc = {};
While[Length[s] > 0, v = Last[s]; 
              done = False; 
While[
                ind[[v]] ≤ 
                    Length[ve[[v]]] && ! done, 
If[! MemberQ[s, (x = ve[[v, ind[[v]]++]])], 
                  done = True]]; 
If[done, AppendTo[s, x], 
                s = Drop[s, -1]; ind[[v]] = 1]; 
              If[(Length[s] == n), 
                AppendTo[NHc, s]; 
If[
                  MemberQ[ve[[x]], 1], 
AppendTo[all, 
                    Append[s, First[s]]]; 
If[
                    c > 0, c--; AppendTo[Hc, s]; 
                    s = Drop[s, -1], 
all = 
                      Flatten[all]; s = {}], s = Drop[s, -1]]]];If[all == {}, 
              If[Length[NHc] == 0, 
                Print["non-Hamiltonian or not connected?"]; Hc = {{}}, 
                Print[{Length[NHc], "non-Hamiltonian Circuits"}]; 
                Hc = NHc]];Hc[[-1]]]

replot[n] creates the n'th graph defined by hamiltonianCycle. If the graph is not Hamiltonian, it creates a near-Hamiltonian graph with a shorter circuit surrounding some "off-circuit" vertices. These can be rotated by an angle a for clarity.
Each replot[n] gives a different embedding, though most are rotations and reflections of each other.

replot[n_:  1, a_:  0]:=   Module[{v = vertx, f, lo, lol, l = Length[vertx], la, m = mx, 
              nn, oldv = vertx},mv = {};new = {};If[Hc == {{}}, Print["No circuit found"]; 
                  Abort[]];f = 
                vertx[[Hc[[n, 1]]]];lo = Hc[[n]];
  
While[
                lo ≠ {} && ! MemberQ[f, (la = Last[lo])], 
                lo = Drop[lo, -1]; PrependTo[mv, la]];
  
ind = {};lol = Length[lo];Do[
                If[i == lo[[i]], , 
                  AppendTo[ind, Rule[lo[[i]], i]]], {i, lol}];
Do[
                If[i == mv[[i]] + lol, , 
                  AppendTo[ind, Rule[mv[[i]], i + lol]]], {i, 
                  Length[mv]}];
  
Which[mv == {}, 
                gS[0, {{1, l, 0}}], 
Length[mv] == 1, 
                gS[0, {{1, l - 1, 0}, {0, 1, 0}}], 
Length[
                    mv] == 4, 
                gS[0, {{1, l - 4, 0}, {0, 1, 0}, { .4, 3, 0, 
                      a}}], 
True, 
                gS[0, {{1, Length[lo], 0}, { .4, Length[mv], 0, a}}]];mx = m;vertx = oldv    ;
  nn = Length[v];
Do[
                ii = i /. ind; 
Do[jj = v[[i, j]]; 
                  jj = jj /. ind; 
                  If[jj > ii, m = Mod[jj - ii, nn]; 
                    AppendTo[
                      new, {{Min[jj, ii], 
                          Max[jj, ii]}}]], 
{j, 
                    Length[v[[i]]]}], 
{i, nn}];
res = Graph[Union[new], comb2]];

Generalized Petersen Graphs.

gP[any__]:=genPetersen[any];genPetersen[o_:  5, s_:  2]:=Which[o == 0 || s == 0 || Abs[s] > Abs[o], 
          Print[ "Bad Data"], 
o < 2 && s == o && 
            OddQ[s],   gS[
            3, {{1, -o, 1}, { .6, -2 o, -s}, { .3, -o, 1, 
                2.8}}, {{-o + 1, 1, 1}}], 
o < 2 && 
            s == o,   gS[
            3, {{1, -o, 1}, { .6, -o, -s}, { .3, -o, 
                1}}, {{-o + 1, 1, 1}}], 
o < -2 && 
            s == -o,   gS[
            3, {{10, 2  s}, {3, 2  s, o,  .4/o}, {7, 2  s, 2, 
                1.3/o}}, {}], 
o < 
              0 && ! EvenQ[s],   gS[
            3, {{1, -o, 1, Mod[o - 1, 2. ] π/o}, { .6, -2 o, 
                s}, { .2, -o, 1, 1.  π}}, {}], 
o > 
              0 && s == o/2,   gS[
            3, {{1. , 2  o}, { .5, o, 
                s, -1.6/o}}, {}], 
o > 0 && 
            s == -o/2,   gS[
            3, {{1. , o}, { .5, o, -s,  .1}}, {{o + 1, 1, 
                1}}], 
True,   gS[
            3, {{1. , Abs[o]}, { .5, Abs[o], s,  .1}}, {}]];

graphNo, regGraphId, cp.

graphNo[ed_:  3, i_, j_:  1, 
          k_:  1]:=Module[{d, gr},gr = 
                Switch[ed, 3, graph3Data, 4, graph4Data, 5, graph5Data, 6, 
                  graph6Data, 7, graph7Data, 8, graph8Data, _, {}];If[Length[gr] < i || Length[gr[[i]]] < j || 
                    Length[(d = gr[[i, j, 3]])] < k,Print[{"Graph not in data ", i, j, k}]; 
                    Abort[]];If[
                Head[d[[k]]] === String, ToExpression[d[[k]]], 
                gS[ed, {{1, i}}, d[[k]]]]];

Clear[regGraphId];vertx =. ;  
regGraphId[gr_, c_:  False]:=Module[{d, ed = Length[vertx[[1]]], chPol, 
            g = "New Graph", i = 0, l, v}, If[c, toVertx[gr]]; 
          ed = Length[vertx[[1]]]; 
          v = Length[vertx]; 
d = 
            ToExpression["graph" <> ToString[ed] <> "Data"]; 
          If[Length[d] < v, 
            Return[g = {"No graphs known of size", ed, v}]]; 
          l = Length[d[[v]]]; 
          If[l < 1, Return[g = {"No graphs known of size", ed, v, l}]]; 
          mx = Length[vertx[[1]]]; 
          chPol = cp[gr]; 
If[MatrixQ[vertx], 
            While[i < l, i++; 
              If[Simplify[chPol == d[[v, i, 2]]], 
                Return[g = {ed, v, i, {d[[v, i, 1]]}}]]], 
            g = "vertx is not a regular graph"]; g]

cp[gr_:  tst]:=Factor[
          CharacteristicPolynomial[ToAdjacencyMatrix[gr], 
              x]/(x - mx)];

Miscellaneous.

EWIsomorphicQ is based on Eric Weisstein's "MyIsomorphicQ" so that the Mathworld Graphs package need not be accessed. As it needs CharacteristicPolynomial from Mathematica5 it fails with earlier versions. The original returned "False" for some graphs with common Hamiltonian Cycle embeddings because "Eccentricity" gave unsorted lists; test has been modified to correct this.
	For regular graphs use the faster "regIsomorphicQ", which omits irrelevant tests.

eccSort[a_]:=Sort[Eccentricity[a]];

EWIsomorphicQ[g_Graph, h_Graph] := Module[{x}, Apply[#1[#2] & , Apply[Equal,And @@ Outer[List, {V, M, Girth, Diameter, DegreeSequence, eccSort}, {g,h}],{1}], {2}] && Subtract @@ (CharacteristicPolynomial[ToAdjacencyMatrix[#1], x] & ) /@ {g, h} === 0]

regIsomorphicQ[g_Graph, h_Graph] := Module[{x}, CharacteristicPolynomial[ToAdjacencyMatrix[g], x] === CharacteristicPolynomial[ToAdjacencyMatrix[h], x]]

toVertx[g_Graph]:= Module[{h,k,l=Length[g[[1]]],r,v=Flatten[g[[1]],1]},r=Table[{},{Max[Flatten[g[[1]]]]}];Do[k=v[[i,1]];h=v[[i,2]];AppendTo[r[[h]],k];AppendTo[r[[k]],h],{i,l}];vertx=r];

gallery[G_:  G6, strt_:  1, 
          ni_:  4, nj_:  8]:=ShowGraphArray[
          Table[G[[strt - 1 + i + ni (j - 1), 1]], {j, nj}, {i, 
              ni}]];

regGraphOptions = {};

SetOptions[ShowGraph, VertexColor -> Red, 
        VertexStyle -> Disk[ .06], VertexNumberColor -> Black, 
        VertexNumberPosition -> { .02, 0},  EdgeColor -> Black, 
        PlotRange -> All];

Graph Data.

This is a collection of regular graph embeddings (mainly as Hamiltonian Cycles, but with some other isomorphic embeddings), taken from the references, and added to at random as I find new incantations.They are stored by the number of vertices and edges, and sorted by decreasing number of automorphisms, with different embeddings indexed. The first element describes the graph as a standard name, or as "Valency, Length, Automorphisms, Index:0". If the index is "hy", the graph is hypersymmetric and the automorphism count has been aborted after an hour or two.
"x=.;" is included to protect the Characteristic Polynomials if the data is modified.

3 regular

The second Heawood embedding, graphNo[3,14,1,2], is a "minimal crossings" version in which the two coloured vertices have been split into two, corresponding to plotting on the surface of a sphere or torus.

x =. ;Clear[graph3Data];graph3Data={{},{},{},{  {"3,4,24", (((1 + x))^3), {{{2, 2}, 2}}}},{},
{  \({"3,6,72", 
                x\^4\ \((3 + x)\), {{{3}, 
                    3}, "gS[3,{{1,4,1,.785},{.25,2}},{{-5,1,1,3,2},{5,4,1,-3,2}}]"}}, {"3,6,12", \((\(-1\) + 
                      x)\)\ x\^2\ \((2 + x)\)\^2, {{{\(-2\), 2, 3}, 1, 
                    1}, "gP[2,1]", "gP[3,1]", "gS[3,{{1.,5},{0,1}},{{3,2,0}}]"}}\)},{},
{  \({"3,8,48", \((\(-1\) + x)\)\^3\ \((1 + x)\)\^3\ \((3 + x)\), {{{\(-3\), 3}, 
                    2}, "gP[4,1]", "gS[3,{{1.,6},{.5,2,1,-.8}},{{2,3,0}}]"}}, 
{"3,8,16,1", \((\(-1\) + x)\)\^2\ \((1 + 
                      x)\)\ \((\(-1\) + 2\ x + x\^2)\)\^2, {{{4}, 
                    4}, {{\(-3\), 3, 4, 4}, 
                    1}, "gP[4,-2]"}}, 
{"3,8,16,2", \((\(-1\) + x)\)\ \((1 + x)\)\^4\ \((\(-5\) + x\^2)\), {{{2, 2}, 
                    3}, "gS[3,{{1,6,1,1.047},{.1,2,0,0,-1}},{{7,1,0,1,3},{8,4,0,1,3}}]"}}, 
{"3,8,12", 
                x\ \((\(-4\) + x + 
                      x\^2)\)\ \((\(-1\) + x + x\^2)\)\^2, {{{\(-3\), \(-3\), 
                      2, 4}, 1, 
                    1}}}, 
{"3,8,4,1", \((\(-1\) + 
                      x)\)\ \((1 + x)\)\^2\ \((\(-3\) + x\^2)\)\ \((\(-1\) + 
                      2\ x + x\^2)\), {{{\(-2\), 4, 2, 4}, 1, 
                    1}, {{\(-2\), 3, \(-3\), 2}, 
                    1}, "gS[3,{{1,7,1,2.02},{0,1,0,-1}},{{1,8,1,0,2},{5,3,0},{3,7,1,-1,2}}]", "gS[3,{{1.,5,1,2.199},{.6,3,1,2.62}},{{3,2,0}}]"}}, 
{"3,8,nx", \((\(-3\) + 
                      x)\)\ \((1 + x)\)\^6, {"gS[3,{{1.,8,2}},{{1,4,-1}}]"}}\)},{},
{  \({"Petersen 120", \((\(-1\) + x)\)\^5\ \((2 + x)\)\^4, {"gS[3,{{1.,9,1},{0,1}},{{3,5,1}}]", "gS[3,{{1,8},{.5,2,1,.3}},{{1,4,2},{-8,10},{-2,9},{2,8,0},{8,1,0}}]", "gP[5,3]", "gS[3,{{1,6},{.5,3,0,-.525},{0,1}},{{-1,7,2,1,3},{1,8,2,1,3},{5,7,0,0,1}}](*[8] P129 ternary symmetry*)", "gS[3,{{1,5,1,0},{.6,4,0,.79},{0.2,1,0}},{{-1,6,4,3,2},{1,10,4,-4,2},{8,10,1,0,2},{6,8,1,1,2},{6,7,0,0,1}}](*[8] P129 ternary symmetry*)"}}, 
{"3,10,48", \((\(-2\) + 
                      x)\)\ \((\(-1\) + x)\)\^2\ x\^2\ \((1 + x)\)\^2\ \((2 + 
                      x)\)\ \((3 + x)\), {{{5, \(-3\), \(-3\), 3, 3}, 
                    1}}}, 
{"3,10,20,1", \((3 + 
                      x)\)\ \((\(-1\) - x + x\^2)\)\^2\ \((\(-1\) + x + x\^2)\)\^2, {{{5}, 5}, {{\(-3\), 3}, 
                    3}}}, 
{"3,10,20,2", \((\(-1\) + 
                      x)\)\ \((\(-1\) - x + x\^2)\)\^2\ \((1 + 3\ x + x\^2)\)\^2, {{{\(-4\), 4, \(-3\), 5, 3}, 
                    1}, "gP[5,1]", "gS[3,{{1.,6},{.4,4,1,.22}},{{3,3,0}}]"}}, 
{"3,10,16,1", \((\(-1\) + x)\)\^2\ x\ \((1 + x)\)\^2\ \((2 + x)\)\^2\ \((\(-4\) - x + x\^2)\), {{{\(-2\), \(-2\), 5, 2, 2}, 
                    1}}}, 
{"3,10,16,2", 
                x\^2\ \((1 + x)\)\ \((2 + x)\)\ \((\(-1\) - 2\ x + 
                      x\^2)\)\ \((\(-2\) - 3\ x + 2\ x\^2 + x\^3)\), {{{3, 3, 
                      7, 0, 0, 2, 2}, 
                    1}}}, 
{"3,10,12", \((\(-2\) + 
                      x)\)\ \((\(-1\) + x)\)\^3\ \((1 + x)\)\^2\ \((2 + x)\)\^3, {{{\(-4\), \(-2\), 5, 2, 4}, 
                    1}}}, 
{"3,10,8,1", \((\(-1\) + x)\)\^3\ x\ \((1 + x)\)\ \((2 + x)\)\^2\ \((\(-4\) + x + x\^2)\), {{{\(-4\), 
                      4, 5, \(-4\), 4}, 1}, {{5, \(-3\), 4, \(-4\), 3}, 
                    1}, "gS[3,{{1,8},{.2,2,0,.2,6}},{{1,5,2,2,2}}]", "gS[3,{{1,9},{0,1}},{{2,5,4,4,2},{3,5,0}}]", "gS[3,{{1,5,1,-.315},{.5,5,2,-.315}},{{-6,9,2,1,2},{6,10,2,-1,2}}]" (*Harary\ book\ cover*) }}, 
{"3,10,8,2", \((\(-1\) - 
                      x + x\^2)\)\ \((\(-1\) + x + x\^2)\)\^2\ \((\(-7\) - 
                      4\ x + 2\ x\^2 + x\^3)\), {{{\(-2\), 5, 5, 2}, 5, 
                    1}}}, 
{"3,10,8,3", 
                x\^2\ \((1 + x)\)\ \((2 + x)\)\^2\ \((\(-3\) + 
                      x\^2)\)\ \((\(-1\) - 2\ x + x\^2)\), {{{\(-2\), 
                      2, \(-3\), 0, 2, 2}, 1, 
                    4}}}, 
{"3,10,6,1", \((\(-1\) + 
                      x)\)\ \((2 + x)\)\^2\ \((\(-1\) - 3\ x + x\^3)\)\^2, {{{\(-2\), 5, 2, \(-4\), 0, 2}, 
                    1, \(-1\)}, "gS[3,{{1.,9},{.0,1}},{{3,2,1}}]", "gS[3,{{1.,7,1,.225},{.3,3,1,.53}},{{4,2,1}}]"}}, 
{"3,10,6,2", \((\(-1\) - 5\ x + x\^2 + 
                      x\^3)\)\ \((\(-1\) - 2\ x + x\^2 + x\^3)\)\^2, {{{\(-2\), 5, 3, \(-4\), 3}, 
                    1, \(-3\)}}}, 
{"3,10,4,1", \((\(-1\) + x)\)\ \((\(-1\) - x + x\^2)\)\ \((\(-3\) + x + x\^2)\)\ \((\(-1\) + x + 
                      x\^2)\)\ \((1 + 3\ x + x\^2)\), {{{\(-4\), 
                      4, \(-4\), \(-4\), 3}, 1, 2}, {{\(-4\), 4, 5, 5, 5}, 1, 
                    2}, {{5, 5, \(-4\), \(-4\), 3}, 2, 
                    2}, "gS[3,{{1,5,1,-.315},{.5,5,1,-.315}},{{-6,10,2,-1,2},{6,9,2,1,2}}]" (*Harary\ book\ cover*) }}, 
{"3,10,4,2", \((\(-1\) - x + x\^2)\)\ \((\(-1\) + x + x\^2)\)\ \((1 + 3\ x + 
                      x\^2)\)\ \((\(-1\) - 4\ x + x\^3)\), {{{\(-2\), \(-4\), 
                      4, 2, 5}, 1, 
                    1}}}, 
{"3,10,4,3", \((1 + 
                      x)\)\ \((\(-1\) - 2\ x + x\^2 + x\^3)\)\ \((1 + 6\ x - 
                      6\ x\^2 - 7\ x\^3 + x\^4 + x\^5)\), {{{\(-2\), \(-2\), 
                      4, 2, 3}, 1}}}, 
{"3,10,4,4", 
                x\ \((\(-1\) - x + 2\ x\^2 + x\^3)\)\ \((4 + 9\ x - 4\ x\^2 - 
                      7\ x\^3 + x\^4 + x\^5)\), {{{\(-3\), \(-2\), 3, 3, 4}, 
                    1}, {{5, \(-2\), \(-4\), 3, 3}, 
                    1}}}, 
{"3,10,2,1", \((\(-1\) + 
                      x)\)\ \((2 + x)\)\ \((\(-1\) - 3\ x + x\^3)\)\ \((2 - 
                      3\ x - 3\ x\^2 + 2\ x\^3 + x\^4)\), {{{\(-2\), 4, 4, 4, 
                      5}, 1, 1}, {{\(-4\), \(-4\), \(-3\), 2, 4}, 
                    1}, {{\(-2\), \(-5\), 3, 4, 5}, 
                    1}, "gS[3,{{1.,7,1,.225},{0.5,3,1,.52}},{{2,4,1}}]"}}, 
{"3,10,2,2", \((\(-1\) + x)\)\ x\ \((1 + x)\)\ \((2 + 
                      x)\)\ \((4 + 10\ x - 5\ x\^2 - 7\ x\^3 + x\^4 + 
                      x\^5)\), {{{\(-2\), 5, \(-3\), 2, 3}, 
                    1, \(-1\)}, {{\(-3\), \(-2\), 4, 2, 4}, 
                    1}, "gS[3,{{1.,8},{.5,2,1,-1.57}},{{3,2,2}}]"}}, 
{"P310 32", \((\(-1\) + 
                      x)\)\ x\^2\ \((1 + x)\)\^2\ \((2 + x)\)\ \((\(-2\) - 
                      7\ x + x\^3)\), {"gS[3,{{1.,8,1,1.965},{.2,2}},{{-1,2,4,4,2},{1,6,0,1,2},{2,5,1,0,2}}]", "gS[3,{{1,5,1,-.32},{.25,4,1,.2},{0,1}},{{1,4,1,1,2}}]"}}\)},{},
{  \({"3,12,288 disconnected", \((\(-3\) + x)\)\^2\ \((\(-1\) + x)\)\^2\ x\^4\ \((2 + x)\)\^4, {"gS[3,{{1,6,4},{.33,6,4}}]"}}, 
{"3,12,64", 
                x\^6\ \((3 + x)\)\ \((\(-4\) - x + x\^2)\)\ \((\(-4\) + x + 
                      x\^2)\), {{{3, 3, 3}, 
                    3}}}, {"Franklin 48", \((\(-1\) + x)\)\^3\ \((1 + x)\)\^3\ \((3 + x)\)\ \((\(-3\) + x\^2)\)\^2, {{{7, 5}, 
                    3}, {{\(-3\), 3, 5, \(-5\)}, 
                    2}, "gP[6,-3]"}}, 
{"3,12,48,2", \((1 + x)\)\^5\ \((3 - 5\ x - x\^2 + x\^3)\)\^2, {{{\(-2\), \(-2\)}, 
                    6}}}, 
{"3,12,36", \((\(-2\) + x)\)\^2\ \((1 + x)\)\^5\ \((\(-3\) + x + x\^2)\)\^2, {{{\(-2\), 6, 2, 6}, 
                    2}}}, 
{"3,12,32", \((\(-1\) + x)\)\^2\ \((1 + x)\)\^5\ \((\(-7\) + x\^2)\)\ \((\(-3\) + 
                      x\^2)\), {{{\(-2\), \(-2\), 6, 6, 2, 2}, 
                    1}}}, 
{"A4 24", \((\(-2\) + x)\)\^3\ x\^2\ \((1 + x)\)\^3\ \((2 + x)\)\^3, {"gS[3,{{1.,6,1,π/6.},{.6,3,0},{.25,3}}]", "gS[3,{{1,9,1,-1.57},{.4,3,1,-1.57}},{{3,5,3,3,2},{2,7,0}}]", "gS[3,{{2.,8,1,π,8.},{1.,4,2}}]"}}, 
{"3,12,24,2", \((\(-2\) + x)\)\^2\ x\^2\ \((1 + x)\)\^3\ \((\(-2\) + 2\ x + x\^2)\)\^2, {{{6}, 
                    6}}}, 
{"3,12,24,3", \((\(-2\) + x)\)\^2\ \((\(-1\) + x)\)\ x\^4\ \((1 + x)\)\ \((2 + x)\)\^2\ \((3 + 
                      x)\), {{{\(-5\), 5, \(-3\), \(-5\), 5, 3}, 
                    1}}}, 
{"3,12,18", \((1 + 
                      x)\)\ \((\(-3\) + x + x\^2)\)\^2\ \((1 - 3\ x + x\^3)\)\^2, {{{6, \(-4\), 5, \(-4\), 4, 5}, 
                    1}}}, 
{"3,12,16,1", \((\(-2\) + 
                      x)\)\ \((\(-1\) + x)\)\^2\ x\^3\ \((2 + x)\)\^3\ \((\(-4\) + x + x\^2)\), {{{\(-5\), 5, 6}, 
                    2}}}, 
{"3,12,16,2", \((\(-1\) + x)\)\^2\ x\ \((2 + x)\)\^2\ \((\(-2\) + x\^2)\)\^2\ \((\(-4\) + x + 
                      x\^2)\), {{{5, 8, 5, 8, 4, 0, 4}, 
                    1}}}, 
{"3,12,16,3", \((\(-1\) + 
                      x)\)\ x\^4\ \((2 + x)\)\ \((\(-4\) + x + 
                      x\^2)\)\ \((\(-4\) - 6\ x + x\^2 + x\^3)\), {{{3, 4, 9, 
                      0, 4, 0, 3, 3}, 
                    1}}}, 
{"3,12,16,4", \((\(-2\) + 
                      x)\)\ \((\(-1\) + x)\)\ x\^3\ \((1 + 
                      x)\)\ \((2 + x)\)\^3\ \((\(-4\) - x + 
                      x\^2)\), {{{\(-2\), 2, \(-3\)}, 
                    3}}}, 
{"3,12,16,5", 
                x\^2\ \((1 + x)\)\^2\ \((6 - 4\ x - 2\ x\^2 + x\^3)\)\ \((2 - 
                      6\ x - 2\ x\^2 + 3\ x\^3 + 
                      x\^4)\), {{{\(-3\), \(-3\), \(-3\), 5, 2, 2}, 
                    1}}}, 
{"3,12,16,6", 
                x\^4\ \((2 + x)\)\ \((\(-4\) - x + x\^2)\)\ \((4 - 6\ x - 
                      5\ x\^2 + 2\ x\^3 + x\^4)\), {{{\(-3\), \(-3\), \(-3\), 
                      2, 3, 0, 2}, 
                    1}}}, 
{"3,12,16,7", \((\(-1\) + 
                      x)\)\ \((1 + x)\)\^3\ \((\(-3\) + x\^2)\)\ \((\(-1\) + 
                      2\ x + x\^2)\)\ \((1 - 5\ x - x\^2 + 
                      x\^3)\), {{{\(-2\), \(-2\), \(-5\), 5, 2, 2}, 
                    1}}}, 
{"3,12,12,1", \((\(-1\) + 
                      x)\)\ x\^2\ \((2 + x)\)\^2\ \((\(-5\) + 
                      x\^2)\)\ \((\(-2\) + x\^2)\)\^2, {"gP[6,2]", "gP[-3,-3]", "gS[3,{{3,3},{1.,6},{3,3,1, 3.15}},{}]", {{\(-4\), \(-2\), 5, \(-5\), 2, 4}, 1}}}, 
{"3,12,12,2", 
                x\^4\ \((3 + x)\)\ \((\(-5\) + 
                      x\^2)\)\ \((\(-2\) + x\^2)\)\^2, {{{\(-3\), \(-5\), \(-3\), 3, 3, 5}, 1}}}, {"3,12,12,3", 
                x\ \((\(-1\) - 2\ x + x\^2 + x\^3)\)\^2\ \((12 - 4\ x - 
                      8\ x\^2 + x\^3 + x\^4)\), {{{\(-5\), \(-3\), \(-3\), 5, 
                      2, 4}, 1}}}, 
{"3,12,12,nh", \((\(-2\) + x)\)\ \((\(-1\) + x)\)\^2\ \((1 + x)\)\^3\ \((2 + 
                      x)\)\ \((\(-3\) + x + x\^2)\)\^2, {"gS[3,{{1,9},{.4,3}},{{0,4,5,0,4},1}]", "gS[3,{{1,11},{0,1,1,1,4}},{{2,11,2,-2,2},{3,7,3,3,2}}]", "gS[3,{{1,7},{0.275,5,1,0,-1}},{{1,8,0,0,1},{3,10,3,1,2},{4,12,1,-3,2},{2,7,0,0,1}}]"}}, 
{"3,12,8,1", \((\(-1\) + x)\)\^2\ \((1 + x)\)\^2\ \((\(-3\) + x\^2)\)\ \((\(-1\) + 2\ x + x\^2)\)\ \((\(-1\) - 
                      5\ x + x\^2 + x\^3)\), {{{\(-5\), 5, \(-4\), \(-4\), 4, 
                      4}, 1}, {{6, 6, \(-4\), \(-4\), 4, 4}, 
                    1}}}, 
{"3,12,8,2", 
                x\^2\ \((1 + x)\)\ \((\(-2\) + x\^2)\)\ \((\(-12\) + 20\ x + 
                      18\ x\^2 - 16\ x\^3 - 9\ x\^4 + 2\ x\^5 + 
                      x\^6)\), {{{\(-2\), \(-2\), 6, 3, 3, 4}, 
                    1}}}, 
{"3,12,8,3", \((\(-1\) + 
                      x)\)\ \((1 + x)\)\^4\ \((3 - 5\ x - x\^2 + 
                      x\^3)\)\ \((1 - 5\ x + x\^2 + 
                      x\^3)\), {{{\(-2\), \(-2\), 5, 3, 5, 3}, 
                    1}}}, 
{"3,12,8,4", \((\(-1\) + 
                      x)\)\ \((\(-3\) + x\^2)\)\ \((\(-1\) + 2\ x + 
                      x\^2)\)\ \((\(-1\) - 3\ x + x\^2 + x\^3)\)\^2, {"gS[3,{{1.,8},{.5,4}},{{2,4,-2}}]"}}, 
{"3,12,8,5", \((\(-2\) + x)\)\ \((\(-1\) + 
                      x)\)\ x\^2\ \((1 + x)\)\ \((2 + x)\)\ \((3 + 
                      x)\)\ \((\(-2\) + x\^2)\)\^2, {{{5, 9, 5, 5, 5, 0, 5}, 
                    1}}}, 
{"3,12,8,6", 
                x\^4\ \((1 + x)\)\ \((5 - 6\ x - x\^2 + x\^3)\)\ \((\(-7\) - 
                      2\ x + 3\ x\^2 + x\^3)\), {{{\(-2\), \(-4\), \(-3\), 3, 
                      3, 3}, 1}}}, 
{"3,12,8,7", \((1 + x)\)\^3\ \((\(-1\) + 2\ x + x\^2)\)\ \((3 - 5\ x - x\^2 + x\^3)\)\ \((1 - 
                      3\ x - x\^2 + x\^3)\), {{{\(-2\), \(-2\), 2, 3, 0, 3, 
                      0, 2}, 1}}}, 
{"3,12,8,8", 
                x\ \((1 + x)\)\^3\ \((2 + x)\)\ \((2 - 4\ x + x\^3)\)\ \((6 - 
                      4\ x - 2\ x\^2 + x\^3)\), {{{\(-2\), 2, \(-3\), 0, 5, 
                      2, 2}, 1}}}, 
{"3,12,8,9", 
                x\^2\ \((1 + x)\)\ \((\(-1\) + x + x\^2)\)\ \((\(-13\) + 
                      25\ x + 31\ x\^2 - 10\ x\^3 - 11\ x\^4 + x\^5 + 
                      x\^6)\), {{{\(-2\), \(-2\), 2, 4, 0, 3, 3}, 
                    1}}}, {"Bidiakis
Cube,8", \((\(-2\) + 
                      x)\)\ x\^4\ \((1 + x)\)\ \((2 + 
                      x)\)\ \((\(-4\) + x + x\^2)\)\^2, {{{\(-4\), 6, 4}, 
                    2}, {{3, 6, 4, \(-4\), 6, \(-3\)}, 
                    2}, {{\(-3\), 6, \(-3\), 3, 6, 3}, 
                    2}, "gS[3,{{.5,7,1,.225},{1,5,1,-.315,1}},{{1,5,0,0,1}}]"}}, 
{"3,12,8,11", \((\(-2\) + 
                      x)\)\ \((1 + x)\)\^5\ \((2 + x)\)\ \((1 - 3\ x + 
                      x\^2)\)\ \((\(-3\) + x + x\^2)\), {{{\(-2\), \(-2\), 2, 
                      6, 0, 2, 2}, 
                    1}}}, 
{"3,12,8,12", \((1 + x)\)\^3\ \((\(-3\) + x\^2)\)\ \((3 - 5\ x - x\^2 + x\^3)\)\ \((\(-1\) - 3\ x + 
                      x\^2 + x\^3)\), {{{\(-2\), 2, \(-5\), 0, \(-5\), 2, 2}, 
                    1}}}, 
{"3,12,8,13", \((\(-1\) + 
                      x)\)\ \((1 + x)\)\^2\ \((\(-1\) + 2\ x + 
                      x\^2)\)\ \((3 - 5\ x - x\^2 + x\^3)\)\ \((\(-1\) - 
                      3\ x + x\^2 + x\^3)\), {{{\(-2\), \(-2\), 4, 5, 3, 4}, 
                    1}}}, 
{"3,12,8,14", \((\(-1\) + 
                      x)\)\ x\^2\ \((1 + x)\)\^2\ \((\(-5\) + 
                      x\^2)\)\ \((\(-5\) + x + x\^2)\)\ \((\(-1\) + x + 
                      x\^2)\), {{{6, \(-2\), \(-5\), \(-5\), 3, 3}, 
                    1}}}, 
{"3,12,4,1", \((\(-2\) + 
                      x)\)\ \((\(-1\) + x)\)\ \((1 + x)\)\^3\ \((2 + 
                      x)\)\ \((\(-3\) + x + x\^2)\)\ \((\(-1\) - 4\ x + 
                      x\^3)\), {{{6, \(-2\), 2, 5, 0, 5, 0, 2}, 1, 
                    3}}}, 
{"3,12,4,2", \((\(-2\) + 
                      x)\)\ \((\(-1\) + x)\)\ x\^2\ \((1 + x)\)\ \((2 + 
                      x)\)\ \((\(-2\) + 2\ x + x\^2)\)\ \((\(-2\) - 4\ x + 
                      x\^2 + x\^3)\), {{{\(-5\), 5, \(-3\), 6, 6, 3}, 
                    1}, 
{{\(-4\), \(-3\), 3, \(-4\), 3, 0, 
                      3}, 1}}}, 
{"3,12,4,3", \((\(-2\) + x)\)\ x\^2\ \((1 + x)\)\^2\ \((\(-2\) + x\^2)\)\ \((\(-4\) + x + 
                      x\^2)\)\ \((\(-2\) + 2\ x + x\^2)\), {{{\(-3\), 
                      4, \(-3\), 5, 3, 0, 4}, 
                    1}}}, 
{"3,12,4,4", 
                x\^4\ \((\(-3\) + 3\ x\^2 + x\^3)\)\ \((17 + x - 9\ x\^2 + 
                      x\^4)\), {{{\(-4\), 3, \(-3\), 2, 0, \(-3\), 3, 3}, 
                    1}}}, 
{"3,12,4,5", \((\(-2\) + 
                      x)\)\ x\^2\ \((1 + x)\)\^2\ \((2 + x)\)\ \((\(-2\) + 
                      2\ x + x\^2)\)\ \((2 - 4\ x - x\^2 + 
                      x\^3)\), {{{\(-3\), \(-2\), 2, 3}, 
                    3}}}, 
{"3,12,4,6", \((1 + 
                      x)\)\ \((\(-3\) + x\^2)\)\ \((\(-1\) + 2\ x + 
                      x\^2)\)\ \((1 - 3\ x - x\^2 + x\^3)\)\ \((\(-1\) - 
                      3\ x + x\^2 + 
                      x\^3)\), {"gS[3,{{4,8,3},{1,4,-1}},{{2,2,2}}]"}}, 
{"3,12,4,7", \((\(-1\) + x)\)\^2\ \((1 + 
                      x)\)\ \((\(-3\) + x\^2)\)\ \((\(-1\) - 3\ x + x\^2 + 
                      x\^3)\)\ \((\(-5\) - x + 3\ x\^2 + 
                      x\^3)\), {"gS[3,{{1,7,1,-.665},{.45,5,2,-.31}},{{4,3,0}}]"}}, 
{"3,12,4,8", \((\(-2\) + x)\)\^2\ x\ \((1 + x)\)\^3\ \((2 + x)\)\ \((\(-2\) + x\^2)\)\ \((\(-2\) + 2\ x + 
                      x\^2)\), {"gS[3,{{1,9,1,-1.57},{.4,3,1,-1.57}},{{3,6,2,2,2},{2,7,0}}]"}}, 
{"3,12,4,9", \((\(-2\) + x)\)\ \((\(-1\) + 
                      x)\)\ \((1 + x)\)\^3\ \((\(-3\) + x + x\^2)\)\ \((2 - 
                      5\ x - 4\ x\^2 + 2\ x\^3 + 
                      x\^4)\), {"gS[3,{{1,9,1,-1.57},{.4,3,1,-1.57}},{{2,5,4,4,2},{3,5,0}}]"}}, 
{"3,12,4,10", 
                x\^2\ \((1 + x)\)\ \((13 - 4\ x - 8\ x\^2 + x\^3 + 
                      x\^4)\)\ \((1 - 4\ x\^2 + x\^3 + 
                      x\^4)\), {{{\(-3\), \(-3\), 4, 4, \(-5\), 3}, 
                    1}}}, 
{"3,12,4,11", 
                x\^2\ \((1 + x)\)\ \((2 + x)\)\^2\ \((5 - 6\ x - x\^2 + 
                      x\^3)\)\ \((1 - 2\ x - x\^2 + 
                      x\^3)\), {{{\(-3\), \(-2\), \(-4\), 2, 3, 0, 2}, 
                    1}}}, 
{"3,12,4,13", 
                x\^2\ \((\(-4\) - 9\ x - 2\ x\^2 + 3\ x\^3 + 
                      x\^4)\)\ \((\(-4\) + 11\ x + x\^2 - 7\ x\^3 + 
                      x\^5)\), {{{\(-2\), 3, 6, 4, 0, 6, 3}, 
                    1}}}, {"3,12,4,14", \((\(-2\) + x)\)\ x\^2\ \((1 + 
                      x)\)\ \((2 + x)\)\ \((\(-2\) + x\^2)\)\ \((2 - 8\ x - 
                      5\ x\^2 + 2\ x\^3 + x\^4)\), {{{\(-4\), \(-2\), 2, 4, 
                      0, 5, 3}, 1}}}, {"3,12,4,15", 
                x\^2\ \((1 + x)\)\^2\ \((2 - 4\ x + x\^3)\)\ \((10 - 6\ x - 
                      8\ x\^2 + x\^3 + 
                      x\^4)\), {{{\(-4\), \(-4\), \(-4\), \(-4\), 2, 2}, 
                    1}}}, 
{"3,12,4,16", \((\(-1\) + 
                      x)\)\ x\ \((1 + x)\)\^3\ \((\(-4\) - x + 
                      x\^2)\)\ \((\(-3\) + x + x\^2)\)\^2, {{{\(-2\), \(-2\), 
                      6, 3, 5, 2}, 
                    1}}}, 
{"3,12,4,17", \((\(-1\) + 
                      x)\)\ x\ \((1 + x)\)\ \((2 + x)\)\ \((\(-2\) + 
                      x\^2)\)\ \((4 + 12\ x - 6\ x\^2 - 8\ x\^3 + x\^4 + 
                      x\^5)\), {{{\(-2\), \(-4\), 4, \(-4\), 4, 2}, 
                    1}}}, 
{"3,12,4,18", 
                x\ \((1 + x)\)\ \((\(-1\) + x + x\^2)\)\ \((\(-16\) - 33\ x + 
                      25\ x\^2 + 35\ x\^3 - 10\ x\^4 - 11\ x\^5 + x\^6 + 
                      x\^7)\), {{{\(-2\), \(-2\), 5, 5, 2, 4}, 
                    1}}}, 
{"3,12,4,19", \((\(-1\) + 
                      x)\)\ x\^2\ \((2 + x)\)\ \((\(-2\) + 
                      x\^2)\)\ \((\(-4\) + x + x\^2)\)\ \((\(-2\) - 4\ x + 
                      x\^2 + x\^3)\), {{{6, \(-5\), \(-3\), 6, 3, 5}, 
                    1}}}, 
{"3,12,4,20", \((\(-1\) + 
                      x + x\^2)\)\ \((\(-1\) - x + 2\ x\^2 + 
                      x\^3)\)\ \((\(-3\) + x + 20\ x\^2 - 9\ x\^4 + 
                      x\^6)\), {{{6, \(-2\), \(-4\), 5, 5, 2}, 
                    1}}}, 
{"3,12,4,21", 
                x\^2\ \((1 + x)\)\ \((13 - 4\ x - 8\ x\^2 + x\^3 + 
                      x\^4)\)\ \((1 - 4\ x - 4\ x\^2 + x\^3 + 
                      x\^4)\), {{{6, \(-3\), \(-3\), 2, 4, 0, 0, 2}, 
                    1}}}, 
{"3,12,4,22", \((\(-1\) + 
                      x)\)\ x\^2\ \((2 + x)\)\^2\ \((\(-2\) + x\^2)\)\ \((2 - 
                      7\ x\^2 + x\^4)\), {{{\(-3\), \(-2\), \(-4\), 4, 2, 3}, 
                    1}}}, 
{"3,12,4,23", \((1 + 
                      x)\)\ \((1 - 3\ x - 2\ x\^2 + 2\ x\^3 + 
                      x\^4)\)\ \((\(-3\) - x + 17\ x\^2 - x\^3 - 9\ x\^4 + 
                      x\^6)\), {{{\(-2\), \(-2\), \(-5\), 4, 2, 3}, 
                    1}}}, 
{"3,12,2,1", \((\(-1\) + 
                      x)\)\ \((3 + 7\ x - 4\ x\^2 - 6\ x\^3 + x\^4 + 
                      x\^5)\)\ \((1 - x - 8\ x\^2 - 2\ x\^3 + 3\ x\^4 + 
                      x\^5)\), {{{\(-4\), 6, 4, 6, \(-5\), 5}, 
                    1}}}, 
{"3,12,2,2", \((1 + 
                      x)\)\ \((\(-3\) + x + x\^2)\)\ \((1 - 3\ x + 
                      x\^3)\)\ \((5 + 6\ x - 6\ x\^2 - 6\ x\^3 + x\^4 + 
                      x\^5)\), {{{\(-2\), 6, \(-5\), 2, 4, 0, 5}, 
                    1}}}, 
{"3,12,2,3", \((\(-1\) + 
                      x)\)\ \((1 + x)\)\ \((1 - 3\ x - 2\ x\^2 + 2\ x\^3 + 
                      x\^4)\)\ \((3 + 16\ x - 5\ x\^2 - 8\ x\^3 + x\^4 + 
                      x\^5)\), {{{6, \(-2\), \(-5\), 4, 6, 3}, 
                    1}}}, 
{"3,12,2,4", \((1 + x)\)\^3\ \((\(-1\) + 7\ x - 4\ x\^2 - 2\ x\^3 + x\^4)\)\ \((3 - 5\ x - 4\ x\^2 + 
                      2\ x\^3 + x\^4)\), {{{\(-2\), \(-4\), 6, \(-4\), 2, 2}, 
                    1}}}, 
{"3,12,2,5", 
                x\^2\ \((3 - 2\ x - 4\ x\^2 + x\^3 + x\^4)\)\ \((7 + 13\ x - 
                      10\ x\^2 - 7\ x\^3 + 2\ x\^4 + x\^5)\), {{{\(-6\), 
                      6, \(-5\), \(-5\), 4, 6}, 
                    1}}}, 
{"3,12,2,6", \((\(-1\) + 
                      x)\)\ x\ \((2 + x)\)\ \((\(-2\) + x\^2)\)\ \((\(-4\) + 
                      8\ x + 14\ x\^2 - 10\ x\^3 - 7\ x\^4 + 2\ x\^5 + 
                      x\^6)\), {{{5, 5, 8, 5, 5, 0, 0, 4}, 
                    1}}}, 
{"3,12,2,7", \((1 + 
                      x)\)\ \((\(-3\) + x + x\^2)\)\ \((1 - 3\ x + 
                      x\^3)\)\ \((1 + 6\ x - 2\ x\^2 - 6\ x\^3 + x\^4 + 
                      x\^5)\), {{{5, 5, 8, 4, 5, 0, 0, 0, 3}, 
                    1}}}, 
{"3,12,2,9", \(\((1 + 
                          x)\)\^2\) \((1 + x - 4\ x\^2 + x\^4)\)\ \((\(-3\) + 
                      14\ x - 3\ x\^2 - 8\ x\^3 + x\^4 + x\^5)\), {{{6, 10, 
                      5, 5, 5, 5}, 1}}}, 
{"3,12,2,10", 
                x\ \((2 + x)\)\ \((2 - x - 4\ x\^2 + x\^3 + 
                      x\^4)\)\ \((\(-4\) + 11\ x + x\^2 - 7\ x\^3 + 
                      x\^5)\), {{{4, 8, 5, 8, 0, 5, 2}, 
                    1}}}, 
{"3,12,2,11", 
                x\^2\ \((13 + 3\ x - 77\ x\^2 - 17\ x\^3 + 82\ x\^4 + 
                      22\ x\^5 - 29\ x\^6 - 9\ x\^7 + 3\ x\^8 + 
                      x\^9)\), {{{\(-2\), \(-4\), 4, \(-4\), 3, 3}, 
                    1}}}, 
{"3,12,2,12", \((\(-2\) + 
                      x)\)\ x\ \((1 + x)\)\^3\ \((2 + x)\)\ \((\(-4\) + 
                      12\ x - 8\ x\^3 + x\^5)\), {{{\(-3\), \(-2\), 2, 7, 0, 
                      2, 2}, 1}}}, 
{"3,12,2,14", 
                x\^2\ \((1 + x)\)\^2\ \((\(-3\) + x + x\^2)\)\ \((\(-7\) + 
                      16\ x + 3\ x\^2 - 9\ x\^3 + x\^5)\), {{{\(-3\), 
                      6, \(-3\), \(-5\), 2, 3}, 
                    1}}}, 
{"3,12,2,15", \((1 + x)\)\^3\ \((\(-3\) + 9\ x - 4\ x\^2 - 2\ x\^3 + x\^4)\)\ \((1 - 7\ x - 4\ x\^2 + 
                      2\ x\^3 + x\^4)\), {{{\(-2\), 6, \(-3\), 2, 5, 0, 2}, 
                    1}}}, 
{"3,12,2,16", \((\(-1\) + 
                      x)\)\ \((1 + x)\)\^3\ \((\(-3\) + x\^2)\)\ \((\(-1\) + 
                      15\ x - 4\ x\^2 - 8\ x\^3 + x\^4 + 
                      x\^5)\), {{{\(-5\), \(-2\), \(-5\), 5, 2, 5}, 
                    1}}}, 
{"3,12,2,17", \((1 + 
                      x)\)\ \((\(-1\) + 7\ x - 4\ x\^2 - 6\ x\^3 + x\^4 + 
                      x\^5)\)\ \((3 + 7\ x - 4\ x\^2 - 6\ x\^3 + x\^4 + 
                      x\^5)\), {{{6, \(-2\), 6, \(-5\), 5, 2}, 
                    1}}}, 
{"3,12,2,18", \((1 + 
                      x)\)\ \((\(-3\) + x + x\^2)\)\ \((1 - 3\ x + 
                      x\^3)\)\ \((1 + 2\ x - 6\ x\^2 - 6\ x\^3 + x\^4 + 
                      x\^5)\), {{{\(-2\), \(-4\), 5, \(-4\), 2, 3}, 
                    1}}}, 
{"3,12,2,19", 
                x\ \((1 + x)\)\ \((\(-1\) - 4\ x + 3\ x\^3 + 
                      x\^4)\)\ \((\(-16\) + 15\ x + 8\ x\^2 - 8\ x\^3 - 
                      x\^4 + x\^5)\), {{{\(-2\), 6, \(-5\), \(-4\), 2, 3}, 
                    1}, "gS[3,{{1,10,1,.629},{0.62,2,0,0,-1}},{{1,11,1,0,2},{8,12,1,0,2},{4,11,1,1,2},{3,7,3,3,2}}]"}}, 
{"3,12,2,20", \((1 + x)\)\ \((\(-2\) + 
                      x\^2)\)\ \((\(-1\) - 2\ x + x\^2 + x\^3)\)\ \((\(-6\) + 
                      16\ x - 3\ x\^2 - 8\ x\^3 + x\^4 + 
                      x\^5)\), {{{\(-5\), \(-3\), 6, \(-4\), 2, 4}, 
                    1}}}, 
{"3,12,2,21", \((1 + x)\)\^2\ \((\(-3\) + 9\ x - 4\ x\^2 - 2\ x\^3 + x\^4)\)\ \((1 + 2\ x - 7\ x\^2 - 
                      2\ x\^3 + 3\ x\^4 + x\^5)\), {{{6, \(-2\), \(-4\), 4, 
                      4, 4}, 1}}}, 
{"3,12,2,22", \((1 + x)\)\ \((\(-1\) + x + x\^2)\)\ \((3 - 9\ x - 38\ x\^2 + 17\ x\^3 + 
                      37\ x\^4 - 8\ x\^5 - 11\ x\^6 + x\^7 + 
                      x\^8)\), {{{\(-5\), \(-2\), \(-5\), \(-5\), 2, 3}, 
                    1}}}, 
{"3,12,1,1", 
                x\^2\ \((2 + x)\)\ \((10 - x - 43\ x\^2 + 10\ x\^3 + 
                      38\ x\^4 - 7\ x\^5 - 11\ x\^6 + x\^7 + 
                      x\^8)\), {{{\(-2\), 6, \(-3\), 3, 4, 4}, 
                    1}} (*First\ of\ 4\ "smallest unimorphic 3-regular"\ graphs*) }, 
{"3,12,1,2", 
                x\^2\ \((1 + x)\)\ \((\(-3\) + x\^2)\)\ \((\(-3\) + x + 
                      x\^2)\)\ \((5 - 3\ x - 6\ x\^2 + x\^3 + 
                      x\^4)\), {{{\(-3\), 3, 5, \(-4\), 0, 5, 2}, 
                    1}}}, 
{"3,12,1,3", 
                x\^2\ \((1 + x)\)\ \((3 - 6\ x - 6\ x\^2 + x\^3 + 
                      x\^4)\)\ \((7 - 2\ x - 6\ x\^2 + x\^3 + 
                      x\^4)\), {{{\(-4\), 3, \(-3\), 2, 0, \(-3\), 4, 2}, 
                    1}, "gS[3,{{1,6},{.5,5,0,.314},{0,1,0,0,-1}},{{9,10,1,1,2},{7,8,0,4,2},{8,12,1,0,2}}]"}}, 
{"3,12,1,4", \((1 + x)\)\ \((3 + 7\ x - 5\ x\^2 - 6\ x\^3 + x\^4 + x\^5)\)\ \((\(-1\) + 7\ x - 
                      x\^2 - 6\ x\^3 + x\^4 + x\^5)\), {{{\(-3\), 6, \(-3\), 
                      5, 2, 5}, 
                    1}, 
{{\(-4\), 6, \(-3\), 6, 2, 5}, 
                    1}, {{\(-5\), \(-5\), \(-3\), 6, 2, 5}, 
                    1}}}, 
{"Frucht ", \((\(-2\) + 
                      x)\)\ x\ \((1 + x)\)\ \((2 + x)\)\ \((\(-1\) - 2\ x + 
                      x\^2 + x\^3)\)\ \((4 - 5\ x - 6\ x\^2 + x\^3 + 
                      x\^4)\), {{{\(-5\), \(-2\), \(-4\), 2, 5, 0, 2}, 
                    1}, "gS[3,{{1.,7,1,π/3.5},{.6,4,0,.78},{0,1}},{{-2,9},{2,6,0},{8,1,0}}]"}}, 
{"P312a", 
                x\^3\ \((1 + x)\)\ \((2 + x)\)\ \((\(-2\) + x\^2)\)\ \((14 - 
                      2\ x - 9\ x\^2 + 
                      x\^4)\), {"gS[3,{{1,7},{0.2,5,1,0,-1}},{{1,8,2,-2,2},{2,4,3,3,2},{9,11,1,1,2}}]"}}, 
{"P312b", 
                x\^4\ \((1 + x)\)\ \((\(-2\) + 2\ x + x\^2)\)\ \((14 - 2\ x - 
                      9\ x\^2 + 
                      x\^4)\), {"gS[3,{{1,7},{0.2,5,1,0,-1}},{{1,8,0,0,1},{2,5,1,1,3},{9,11,1,1,2}}]"}}, 
{"P312c", 
                x\ \((1 + x)\)\^3\ \((\(-1\) + x + x\^2)\)\ \((\(-12\) + 
                      23\ x + 7\ x\^2 - 10\ x\^3 - x\^4 + 
                      
                      x\^5)\), {"gS[3,{{1,7},{0.5,5,1,0,-1}},{{1,8,0,0,1},{2,7,1,1,1},{3,5,1,1,2},{9,11,1,1,2}}]"}}, 
{"P312d", 
                x\^2\ \((1 + x)\)\^2\ \((\(-1\) + x + x\^2)\)\ \((\(-13\) + 
                      26\ x + x\^2 - 11\ x\^3 + 
                      x\^5)\), {"gS[3,{{1,7},{0.2,5,1,0,-1}},{{1,8,0,0,1},{2,6,1,1,1},{3,5,1,2,2},{9,11,1,1,2}}]"}}\)},{},
{  \({"Heawood 336", \((3 + 
                      x)\)\ \((\(-2\) + x\^2)\)\^6, {{{\(-5\), 5}, 
                    7, \(-2\)}, "Graph[{{{1,2}},{{1,3}},{{1,4}},{{2,5}},{{2,11}},{{3,6}},{{3,12}},{{4,7}},{{4,13}},{{5,8}},{{6,8}},{{6,9}},{{7,9}},{{7,10}},{{9,11}},{{10,12}},{{10,15}},{{11,14}},{{12,14}},{{13,14}},{{13,16}},{{15,16}}},{{{0,3}},{{-2,2}},{{0,2}},{{2,2}},{{-3,1},VertexColor->Green},{{-1,1}},{{1,1}},{{-3,-1},VertexColor->Blue},{{-1,-1}},{{1,-1}},{{-2,-2}},{{0,-2}},{{2,-2}},{{0,-3}},{{3,1},VertexColor->Green},{{3,-1},VertexColor->Blue}}]"}}, 
{"3,14,32,1", \((\(-1\) + 
                      x)\)\ x\^2\ \((1 + x)\)\^3\ \((\(-4\) - x + 
                      x\^2)\)\ \((\(-1\) + 2\ x + x\^2)\)\ \((4 - 7\ x + 
                      x\^3)\), {{{\(-3\), \(-3\), 3, 3, \(-5\), 0, 0, 2, 2}, 
                    1}}}, 
{"3,14,32,2", \((\(-1\) + 
                      x)\)\ x\^2\ \((1 + x)\)\^2\ \((\(-2\) - 3\ x + 
                      2\ x\^2 + x\^3)\)\ \((10 + 21\ x - 2\ x\^2 - 10\ x\^3 + 
                      x\^5)\), {{{6, 2, 2, 0, 0, 8, 8, 3, 3, 3}, 
                    1}}}, 
{"3,14,28,1", \((3 + 
                      x)\)\ \((1 - x - 2\ x\^2 + x\^3)\)\^2\ \((\(-1\) - x + 2\ x\^2 + x\^3)\)\^2, {{{7}, 7}, {{\(-3\), \(-5\), 7, 5, 3, 7, 7}, 
                    1, \(-2\)}, {{\(-3\), 3}, 
                    7}}}, 
{"3,14,28,2", \((\(-1\) + 
                      x)\)\ \((1 - x - 2\ x\^2 + x\^3)\)\^2\ \((\(-1\) + 3\ x + 4\ x\^2 + x\^3)\)\^2, {{{\(-3\), \(-5\), 7, 5, 3, \(-6\), 6}, 1, 
                    5}}}, 
{"3,14,24", \((\(-2\) + 
                      x)\)\ x\ \((1 + x)\)\^2\ \((\(-4\) - 6\ x + x\^2 + 
                      x\^3)\)\ \((\(-1\) - 3\ x + x\^2 + x\^3)\)\^2, {{{6, 2, 
                      6, 0, 7, 2, 8, 0, 0, 3, 3}, 1, 
                    3}}}, 
{"3,14,16,1", \((\(-1\) + 
                      x)\)\ \((1 + x)\)\^2\ \((\(-1\) - x + x\^2)\)\ \((1 + 
                      3\ x + x\^2)\)\ \((\(-3\) - 6\ x + x\^3)\)\ \((\(-1\) - 
                      4\ x + x\^3)\), {{{7, 2, 2, 0, 0, 3, 7, 7, 0, 2, 2}, 
                    1}}}, 
 (*\({"3,14,16,2", 
                  x\ \((1 + x)\)\^2\ \((\(-4\) - 6\ x + x\^2 + 
                        x\^3)\)\ \((\(-1\) - 3\ x + x\^2 + x\^3)\)\ \((2 + 
                        x - 5\ x\^2 - x\^3 + x\^4)\), {{{7, 2, 2, 0, 0, 3, 7, 
                        7, 0, 2, 2}, 
                      1}}}\)\(,\)*) 
{"3,14,16,3", \((1 + x)\)\^2\ \((\(-1\) - x + x\^2)\)\ \((\(-1\) + x + x\^2)\)\ \((\(-1\) - 
                      4\ x + x\^3)\)\ \((5 - 7\ x - 8\ x\^2 + x\^3 + 
                      x\^4)\), {{{6, 2, 2, 0, 0, 3, 8, 6, 0, 2, 2}, 
                    1}}}, 
{"3,14,16,4", \((\(-1\) + 
                      x)\)\ x\^2\ \((1 + x)\)\^2\ \((\(-3\) + 
                      x\^2)\)\ \((\(-4\) - x + x\^2)\)\ \((\(-4\) + x + 
                      x\^2)\)\ \((\(-1\) + 2\ x + x\^2)\), {{{8, 2, 2, 0, 0, 
                      6, 3, 5, 6, 0, 3}, 
                    1}}}, 
{"3,14,16,5", \((\(-1\) + 
                      x)\)\ x\^2\ \((1 + x)\)\^2\ \((2 + x)\)\ \((\(-3\) + 
                      x\^2)\)\ \((10 + 21\ x - 2\ x\^2 - 10\ x\^3 + 
                      x\^5)\), {{{6, 2, 2, 0, 0, 8, 8, 2, 3, 0, 2}, 
                    1}}}, 
{"3,14,16,6", \((\(-1\) + x)\)\^2\ x\ \((1 + x)\)\^3\ \((2 + x)\)\ \((\(-28\) + 10\ x + 39\ x\^2 - 
                      2\ x\^3 - 12\ x\^4 + x\^6)\), {{{6, 2, 2, 0, 0, 8, 8, 
                      5, 2, 2}, 1}}}, 
{"3,14,16,7", 
                x\^2\ \((1 + x)\)\ \((\(-4\) + 9\ x + 8\ x\^2 - 6\ x\^3 - 
                      2\ x\^4 + x\^5)\)\ \((4 - x - 10\ x\^2 + 4\ x\^4 + 
                      x\^5)\), {{{5, 2, 2, 0, 0, 9, 7, 3, 3, 3}, 
                    1}}}, 
{"3,14,16,8", \((\(-1\) + x)\)\^2\ \((1 + x)\)\^2\ \((\(-5\) - x + x\^2)\)\ \((\(-1\) + x + 
                      x\^2)\)\ \((1 + 3\ x + x\^2)\)\ \((\(-1\) - 4\ x + 
                      x\^3)\), {{{5, 2, 2, 0, 0, 9, 7, 5, 2, 2}, 
                    1}}}, 
{"3,14,16,9", 
                x\^2\ \((2 + x)\)\^2\ \((\(-2\) + x\^2)\)\^2\ \((\(-2\) - 
                      2\ x + x\^2)\)\ \((\(-2\) - 4\ x + x\^2 + 
                      x\^3)\), {{{7, 2, 3, 0, 2, 0, 0, 7, 2, 3, 0, 2}, 
                    1}}}, 
{"3,14,16,10", \((\(-1\) + 
                      x)\)\ x\^3\ \((2 + x)\)\^2\ \((\(-2\) + 
                      x\^2)\)\ \((\(-2\) - 2\ x + x\^2)\)\ \((\(-4\) - 4\ x + 
                      2\ x\^2 + x\^3)\), {{{7, 2, 3, 0, 2, 0, 0, 7, 3, 3, 3}, 
                    1}}}, 
{"3,14,16,11", 
                x\^2\ \((1 + x)\)\ \((\(-1\) + 2\ x + x\^2)\)\ \((2 + x - 
                      5\ x\^2 - x\^3 + x\^4)\)\ \((6 - 3\ x - 7\ x\^2 + 
                      x\^3 + x\^4)\), {{{11, 2, 3, 0, 3, 0, 2, 0, 0, 3, 3, 
                      3}, 1}}}, 
{"3,14,16,12", 
                x\^3\ \((1 + x)\)\ \((\(-1\) + x + x\^2)\)\ \((16 - 56\ x + 
                      12\ x\^2 + 57\ x\^3 - 9\ x\^4 - 14\ x\^5 + x\^6 + 
                      x\^7)\), {{{11, 2, 4, 0, 3, 3, 0, 0, 0, 3, 3, 3}, 
                    1}}}, 
{"3,14,16,13", \((\(-1\) + x)\)\^2\ \((1 + x)\)\ \((\(-1\) + x + x\^2)\)\^2\ \((\(-1\) - 4\ x + 
                      x\^3)\)\ \((\(-11\) - 6\ x + 2\ x\^2 + x\^3)\), {{{12, 
                      2, 4, 0, 3, 5, 0, 0, 3, 4, 0, 0, 2}, 
                    1}}}, 
{"3,14,16,14", 
                x\^2\ \((1 + x)\)\ \((2 + x)\)\ \((\(-3\) + x\^2)\)\ \((1 - 
                      5\ x + x\^2 + x\^3)\)\ \((2 + x - 5\ x\^2 - x\^3 + 
                      x\^4)\), {{{5, 2, 5, 0, 2, 9, 0, 0, 3, 3, 3}, 
                    1}}}, 
{"3,14,14", \((\(-1\) + 
                      x)\)\ \((\(-1\) + 11\ x + 10\ x\^2 - 10\ x\^3 - 6\ x\^4 + 2\ x\^5 + x\^6)\)\^2, {{{\(-4\), 5, 6, \(-4\), 5, 7, 0, 4}, 1, 
                    2}, "gP[7,2]", "gP[7,3]", "gS[3,{{1,7,2},{0.3,7,-1}}]"}}, 
{"3,14,12,1", \((\(-1\) + x)\)\ x\^2\ \((2 + x)\)\^2\ \((\(-6\) + 
                      x\^2)\)\ \((\(-2\) + x\^2)\)\^3, {{{5, 2, 4, 0, 5, 9, 
                      0, 4, 5, 0, 2}, 1, 
                    4}}}, 
{"3,14,12,2", 
                x\ \((2 + x)\)\ \((\(-4\) - 6\ x + x\^2 + 
                      x\^3)\)\ \((1 - x - 4\ x\^2 + x\^4)\)\^2, {{{11, 2, 6, 
                      0, 2, 7, 0, 2, 0, 0, 3, \(-3\)}, 1, 
                    2}}}, 
{"3,14,12,3", \((1 + x)\)\^2\ \((\(-3\) + x + x\^2)\)\ \((\(-1\) - 4\ x + x\^3)\)\^3, {{{12, 2, 6, 0, 6, 
                      2, 7, 0, 0, 2, 0, 0, 2}, 1, 
                    1}}}, 
{"3,14,8,2", 
                x\ \((1 + x)\)\ \((2 + x)\)\ \((\(-1\) + 2\ x + 
                      x\^2)\)\ \((2 + x - 5\ x\^2 - x\^3 + x\^4)\)\ \((4 + 
                      3\ x - 5\ x\^2 - x\^3 + x\^4)\), {{{\(-3\), \(-2\), 2}, 
                    5, \(-1\)}}}, 
{"3,14,8,3", \((\(-2\) + x)\)\ \((\(-1\) + x)\)\^2\ x\^2\ \((1 + x)\)\ \((2 + x)\)\ \((\(-5\) + 
                      x\^2)\)\ \((\(-1\) + 2\ x + x\^2)\)\^2, {{{7, \(-6\), 
                      6}, 3, \(-1\)}}}, 
{"3,14,8,4", \((\(-1\) + x)\)\^2\ x\ \((1 + x)\)\ \((2 + x)\)\ \((2 + x - 5\ x\^2 - x\^3 + 
                      x\^4)\)\ \((\(-4\) - 11\ x - 3\ x\^2 + 3\ x\^3 + 
                      x\^4)\), {{{\(-3\), \(-5\), 2}, 
                    5, \(-1\)}}}, 
{"3,14,8,5", 
                x\ \((2 + x)\)\ \((\(-2\) + 
                      x\^2)\)\ \((\(-2\) - 4\ x + x\^3)\)\^2\ \((\(-2\) - 
                      4\ x + x\^2 + x\^3)\), {{{\(-2\), 2, 7, 0, 7, 2, 7, 0, 
                      2}, 1, \(-4\)}}}, 
{"3,14,8,6", \((\(-1\) + x)\)\^2\ \((1 + x)\)\ \((\(-1\) + x + x\^2)\)\ \((1 + 3\ x + 
                      x\^2)\)\ \((3 - 6\ x + x\^3)\)\ \((\(-1\) - 4\ x + 
                      x\^3)\), {{{7, 7, \(-6\), 6, \(-6\), 6, 7}, 
                    1}}}, 
{"3,14,8,7", \((\(-2\) + x\^2)\)\^4\ \((\(-2\) + 2\ x + x\^2)\)\ \((\(-2\) - 4\ x + x\^2 + 
                      x\^3)\), {"gS[3,{{1,9,1,1.922},{.5,5,2,2.19}},{{3,7,2,2,2}}]"}}, 
{"3,14,8,8", \((\(-1\) + x)\)\ \((1 + x)\)\ \((3 + 
                      x)\)\ \((\(-1\) - x + x\^2)\)\ \((\(-1\) + x + 
                      x\^2)\)\ \((\(-1\) - 4\ x + x\^3)\)\ \((1 - 4\ x + 
                      x\^3)\), {{{7, 11, 7, 5, 7, 5, 7}, 
                    1}}}, 
{"3,14,6", \((1 + x)\)\^3\ \((\(-3\) + x\^2)\)\ \((\(-1\) + 2\ x + 
                      x\^2)\)\ \((3 - 4\ x - x\^2 + x\^3)\)\^2, {{{8, 2, 4, 
                      0, 8, 5, 0, 2, 6, 0, 0, 2}, 
                    1}}}, 
{"3,14,4,1", \((1 - 3\ x + 
                      x\^3)\)\ \((1 - x - 2\ x\^2 + x\^3)\)\ \((\(-1\) - x + 
                      2\ x\^2 + x\^3)\)\ \((1 - 10\ x - 3\ x\^2 + 3\ x\^3 + 
                      x\^4)\), {{{\(-6\), \(-3\), 4}, 
                    5, \(-2\)}, {{7, 7, 7, \(-4\), 7, 7, 4}, 
                    1, \(-1\)}}}, 
{"3,14,4,2", \((3 + 
                      x)\)\ \((\(-1\) - 3\ x + x\^3)\)\ \((1 - 3\ x + 
                      x\^3)\)\ \((1 - x - 2\ x\^2 + x\^3)\)\ \((\(-1\) - x + 
                      2\ x\^2 + x\^3)\), {{{\(-3\), 7, 7, 7, 3}, 
                    3, \(-2\)}, {{7, \(-5\), 7, 5, 7, 7, 7}, 1, 
                    5}}}, 
{"3,14,4,3", \((\(-1\) + 
                      x)\)\ x\ \((1 - x - 3\ x\^2 + x\^3 + 
                      x\^4)\)\ \((\(-4\) + 9\ x + 53\ x\^2 + 12\ x\^3 - 
                      26\ x\^4 - 8\ x\^5 + 3\ x\^6 + x\^7)\), {{{\(-4\), 
                      5, \(-5\), \(-4\), 3}, 
                    4}}}, 
{"3,14,4,4", \((\(-1\) + 
                      x)\)\ \((\(-1\) + x + x\^2)\)\ \((\(-1\) - 4\ x + 
                      x\^3)\)\ \((1 - 4\ x + x\^3)\)\ \((1 - 7\ x - 2\ x\^2 + 
                      3\ x\^3 + x\^4)\), {{{7, \(-4\), \(-7\), \(-5\), 6, 3}, 
                    2}}}, 
{"3,14,4,5", 
                x\^2\ \((3 + x)\)\ \((\(-2\) + x\^2)\)\^2\ \((\(-2\) - 4\ x + 
                      x\^3)\)\ \((2 - 4\ x + x\^3)\), {{{7, \(-5\), 
                      3, \(-5\)}, 
                    4, \(-2\)}, {{\(-3\), 7, \(-3\), 7, 3, 7, 3}, 
                    1, \(-3\)}}}, 
{"3,14,4,6", \((1 + 
                      x)\)\ \((\(-2\) + x\^2)\)\ \((1 - 2\ x - 6\ x\^2 + 
                      x\^4)\)\ \((\(-2\) + 4\ x + 7\ x\^2 - 8\ x\^3 - 
                      6\ x\^4 + 2\ x\^5 + x\^6)\), {{{\(-3\), \(-5\), 
                      7, \(-5\), \(-5\), 2, 2}, 1, 
                    2}}}, 
{"3,14,4,7", 
                x\ \((1 - x - 3\ x\^2 + x\^3 + x\^4)\)\ \((4 - 13\ x - 
                      24\ x\^2 + 37\ x\^3 + 34\ x\^4 - 18\ x\^5 - 11\ x\^6 + 
                      2\ x\^7 + x\^8)\), {{{\(-3\), 6, \(-3\), \(-5\), 6, 3, 
                      3}, 1, 3}}}, 
{"3,14,4,8", \((\(-1\) + x)\)\ \((\(-3\) - 5\ x + x\^3)\)\ \((1 - x - 2\ x\^2 + 
                      x\^3)\)\ \((\(-1\) - x + 2\ x\^2 + x\^3)\)\ \((\(-1\) + 
                      3\ x + 4\ x\^2 + x\^3)\), {{{\(-3\), \(-2\), 2, 3, 0, 
                      3, 0, 3, 0, 3}, 
                    1}}}, 
{"3,14,4,9", \((\(-2\) + x\^2)\)\^2\ \((\(-2\) - 4\ x + x\^3)\)\ \((\(-2\) - 4\ x + x\^2 + 
                      x\^3)\)\ \((\(-2\) - 2\ x + 2\ x\^2 + 
                      x\^3)\), {"gS[3,{{1,9,1,1.922},{.5,5,2,2.19}},{{3,5,4,4,2}}]"}}, 
{"3,14,4,10", \((1 - x - 2\ x\^2 + 
                      x\^3)\)\ \((\(-3\) - 3\ x + 2\ x\^2 + 
                      x\^3)\)\ \((\(-1\) - x + 2\ x\^2 + x\^3)\)\ \((1 - 
                      2\ x - 5\ x\^2 + x\^3 + x\^4)\), {{{7, 7, 11, 6, 6, 6, 
                      6}, 1}}}, 
{"3,14,4,11", \((\(-1\) + x)\)\ \((1 - 3\ x + x\^3)\)\ \((1 - x - 2\ x\^2 + 
                      x\^3)\)\ \((\(-5\) - 3\ x + 2\ x\^2 + 
                      x\^3)\)\ \((\(-1\) + 3\ x + 4\ x\^2 + x\^3)\), {{{7, 9, 
                      7, 5, 7, 8, 6}, 
                    1}}}, 
{"3,14,2,1", \((1 + 
                      x)\)\ \((\(-1\) - x + x\^2)\)\ \((\(-1\) - 4\ x + 
                      x\^3)\)\ \((\(-5\) + 3\ x + 30\ x\^2 + 6\ x\^3 - 
                      20\ x\^4 - 6\ x\^5 + 3\ x\^6 + x\^7)\), {{{\(-2\), 7, 
                      3}, 5}}}, 
{"3,14,2,2", \((\(-1\) + x)\)\ \((\(-5\) + 11\ x + 10\ x\^2 - 10\ x\^3 - 6\ x\^4 + 2\ x\^5 + 
                      x\^6)\)\ \((\(-1\) + 11\ x + 10\ x\^2 - 10\ x\^3 - 
                      6\ x\^4 + 2\ x\^5 + x\^6)\), {{{\(-4\), 5, 7, \(-6\), 
                      4, \(-6\), \(-5\), 5}, 
                    1}}}, 
{"3,14,2,3", \((\(-3\) + 
                      10\ x + 9\ x\^2 - 10\ x\^3 - 6\ x\^4 + 2\ x\^5 + 
                      x\^6)\)\ \((\(-1\) - 5\ x + 5\ x\^2 + 15\ x\^3 - 
                      4\ x\^4 - 8\ x\^5 + x\^6 + x\^7)\), {{{\(-6\), \(-3\), 
                      5, 7, 7, \(-6\), 3}, 
                    1}}}, 
{"3,14,2,4", 
                x\ \((\(-2\) + x\^2)\)\ \((2 - 4\ x + x\^3)\)\ \((\(-4\) + 
                      28\ x\^2 + 6\ x\^3 - 20\ x\^4 - 6\ x\^5 + 3\ x\^6 + 
                      x\^7)\), {{{\(-5\), 6, 6, 7, \(-5\), 6, 6}, 
                    1, \(-3\)}}}, 
{"3,14,2,5", \((5 + 
                      15\ x + 6\ x\^2 - 12\ x\^3 - 6\ x\^4 + 2\ x\^5 + 
                      x\^6)\)\ \((\(-1\) - 2\ x + 9\ x\^2 + 14\ x\^3 - 
                      6\ x\^4 - 8\ x\^5 + x\^6 + x\^7)\), {{{7, \(-4\), 2, 6, 
                      4}, 5, 1}}}, 
{"3,14,2,6", \((\(-1\) + x)\)\ \((1 + x)\)\ \((2 + x)\)\ \((2 + 23\ x + 29\ x\^2 - 
                      78\ x\^3 - 82\ x\^4 + 55\ x\^5 + 55\ x\^6 - 13\ x\^7 - 
                      13\ x\^8 + x\^9 + x\^10)\), {{{\(-2\), \(-4\), \(-5\), 
                      6}, 3}}}, 
{"3,14,2,7", \((\(-2\) + x\^2)\)\ \((\(-1\) + 6\ x + 5\ x\^2 - 6\ x\^3 - x\^4 + x\^5)\)\ \((2 + 
                      6\ x - 7\ x\^2 - 13\ x\^3 + 4\ x\^5 + 
                      x\^6)\), {{{\(-5\), 4, \(-6\), \(-6\), \(-5\), 0, 2, 
                      5}, 1}}}, 
{"3,14,2,8", \((\(-2\) + x\^2)\)\ \((1 - 4\ x\^2 + x\^4)\)\ \((\(-2\) - 14\ x + 27\ x\^2 + 
                      13\ x\^3 - 18\ x\^4 - 6\ x\^5 + 3\ x\^6 + 
                      x\^7)\), {{{\(-6\), 5, 7, \(-4\)}, 
                    3, \(-2\)}}}, 
{"3,14,2,9", 
                x\ \((2 + x)\)\ \((\(-2\) + x\^2)\)\^2\ \((\(-2\) - 4\ x + 
                      x\^3)\)\ \((2 - 2\ x - 6\ x\^2 + x\^3 + 
                      x\^4)\), {{{\(-5\), \(-2\), 2, 5, 3, 5, 5, 5}, 1, 
                    2}}}, 
{"3,14,2,10", \((\(-1\) + 
                      x)\)\ x\ \((3 - 8\ x - 13\ x\^2 + 4\ x\^4 + 
                      x\^5)\)\ \((\(-4\) + 3\ x + 16\ x\^2 - x\^3 - 8\ x\^4 + 
                      x\^6)\), {"gS[3,{{1,9,1,1.922},{.4,5,1,2.19}},{{3,9,2,-2,2}}]"}}, 
{"3,14,2,11", \((\(-1\) + 3\ x + 11\ x\^2 - 8\ x\^3 - 
                      6\ x\^4 + 2\ x\^5 + x\^6)\)\ \((1 - 12\ x + 8\ x\^2 + 
                      19\ x\^3 - 6\ x\^4 - 8\ x\^5 + x\^6 + 
                      x\^7)\), {"gS[3,{{1,9,1,1.922},{.5,5,1,2.19}},{{3,7,2,2,2}}]"}}, 
{"3,14,2,12", \((\(-2\) + x\^2)\)\^2\ \((\(-3\) - 
                      5\ x + 2\ x\^2 + 4\ x\^3 + x\^4)\)\ \((\(-1\) + 6\ x + 
                      5\ x\^2 - 6\ x\^3 - x\^4 + 
                      x\^5)\), {"gS[3,{{1,9,1,1.922},{.5,5,2,2.19}},{{3,9,2,-2,2}}]"}}, 
{"3,14,2,13", \((1 + x)\)\ \((\(-1\) + 8\ x - 
                      5\ x\^2 - 7\ x\^3 + x\^4 + x\^5)\)\ \((1 - 12\ x + 
                      8\ x\^2 + 19\ x\^3 - 6\ x\^4 - 8\ x\^5 + x\^6 + 
                      x\^7)\), {"gS[3,{{1,9,1,1.922},{.5,5,1,2.19}},{{3,5,4,4,2}}]"}}, 
{"3,14,2,14", \((\(-1\) + 
                      x)\)\ \((\(-1\) + 2\ x + x\^2)\)\^2\ \((\(-1\) + 6\ x - 
                      3\ x\^2 - 2\ x\^3 + x\^4)\)\ \((\(-1\) - 6\ x - 
                      3\ x\^2 + 2\ x\^3 + x\^4)\), {{{7, 8, 6, 7, 9, 6, 6}, 
                    1}}}, 
{"3,14,2,15", \((\(-3\) + 
                      3\ x + 10\ x\^2 - 8\ x\^3 - 6\ x\^4 + 2\ x\^5 + 
                      x\^6)\)\ \((\(-1\) - 2\ x + 9\ x\^2 + 14\ x\^3 - 
                      6\ x\^4 - 8\ x\^5 + x\^6 + x\^7)\), {{{7, 7, 7, 10, 6, 
                      6, 6}, 1}}}, 
{"3,14,1,1", \((\(-1\) + x + x\^2)\)\ \((\(-3\) - 2\ x + 52\ x\^2 + 55\ x\^3 - 122\ x\^4 - 
                      100\ x\^5 + 87\ x\^6 + 58\ x\^7 - 23\ x\^8 - 13\ x\^9 + 
                      2\ x\^10 + x\^11)\), {{{\(-2\), 6, 6, 6, 7, 5, 7}, 1, 
                    1}}}, 
{"3,14,1,2", 
                x\ \((\(-1\) - x + x\^2)\)\ \((16 + 13\ x - 104\ x\^2 - 
                      100\ x\^3 + 107\ x\^4 + 133\ x\^5 - 8\ x\^6 - 
                      43\ x\^7 - 7\ x\^8 + 4\ x\^9 + 
                      x\^10)\), {{{\(-3\), \(-2\), 6, 6, 6, 2, 6}, 
                    1}}}, 
{"3,14,1,3", 
                4 - 32\ x - 89\ x\^2 + 123\ x\^3 + 345\ x\^4 - 67\ x\^5 - 
                  400\ x\^6 - 46\ x\^7 + 192\ x\^8 + 46\ x\^9 - 40\ x\^10 - 
                  12\ x\^11 + 3\ x\^12 + 
                  x\^13, {{{\(-2\), \(-3\), 6, \(-2\), 6, 8, 3, 4}, 
                    1, \(-2\)}}}, 
{"3,14,1,4", 
                3 - 5\ x - 69\ x\^2 - 15\ x\^3 + 245\ x\^4 + 97\ x\^5 - 
                  298\ x\^6 - 112\ x\^7 + 156\ x\^8 + 54\ x\^9 - 36\ x\^10 - 
                  12\ x\^11 + 3\ x\^12 + 
                  x\^13, {{{\(-3\), 4, 7, 5, \(-6\), \(-4\), 7, 3}, 
                    1, \(-2\)}}}, 
{"3,14,1,5", \(-1\) + 17\ x - 38\ x\^2 - 80\ x\^3 + 205\ x\^4 + 103\ x\^5 - 289\ x\^6 - 
                  89\ x\^7 + 162\ x\^8 + 48\ x\^9 - 38\ x\^10 - 12\ x\^11 + 
                  3\ x\^12 + 
                  x\^13, {{{\(-2\), \(-5\), \(-3\), 5, 4}, 
                    4}}}, 
{"3,14,1,6", 
                3 - 5\ x - 43\ x\^2 + 33\ x\^3 + 184\ x\^4 - 44\ x\^5 - 
                  286\ x\^6 - 20\ x\^7 + 174\ x\^8 + 40\ x\^9 - 40\ x\^10 - 
                  12\ x\^11 + 3\ x\^12 + 
                  x\^13, {{{10, 2, 3, 0, 3, 0, 3, 0, 4, 0, 4, 2}, 
                    1}}}, 
{"3,14,1,7", \((\(-2\) + x\^2)\)\^2\ \((3 + 5\ x - 40\ x\^2 - 24\ x\^3 + 58\ x\^4 + 22\ x\^5 - 
                      24\ x\^6 - 8\ x\^7 + 3\ x\^8 + x\^9)\), {{{6, 10, 6, 6, 
                      9, 5, 0, 5}, 
                    1}}}, 
{"3,14,44nh", \((\(-1\) + 
                      x)\)\ \((1 + x)\)\^2\ \((\(-3\) + x + x\^2)\)\^2\ \((\(-1\) - 4\ x + x\^3)\)\^2, {"gS[3,{{1,11,1,1.85},{.2,3,1,2.62,10}},{{1,5,1,4,2},{4,10,2,-2,2}}]"}}, 
{"P314a 64", \((\(-1\) + 
                      x)\)\ x\^2\ \((1 + x)\)\^3\ \((\(-4\) - x + 
                      x\^2)\)\ \((\(-1\) + 2\ x + x\^2)\)\ \((\(-4\) - 7\ x + 
                      x\^3)\), {"gS[3,{{1.,10,1,1.88},{.2,4,1,0,-1}},{{-1,2,5,5,2},{1,7,1,-1,2},{2,5,1,1,2},{4,11,5,2,2},{1,8,6,2,2},{12,2,0}}]"}}, 
{"P314b 128", x\^4\ \((\(-2\) + x\^2)\)\ \((\(-2\) + 2\ x + x\^2)\)\^2\ \((2 - 
                      6\ x - x\^2 + 
                      x\^3)\), {"gS[3,{{1.,12,1,1.35},{.1,2,1}},{{-1,12,6,-6,2},{1,3,6,6,2},{1,4,6,6,2},{1,6,6,6,2},{2,4,6,6,2},{2,5,6,6,2},{2,6,6,6,2},{3,6,6,6,2}}]"}}\)},{},
{  \({"3,16,128", \((\(-1\) + x)\)\^5\ \((1 + x)\)\^5\ \((3 + x)\)\ \((\(-5\) + x\^2)\)\^2, {{{7, \(-7\)}, 
                    6}}}, 
{"MoebiusKantor 96", \((\(-1\) + x)\)\^3\ \((1 + x)\)\^3\ \((3 + 
                      x)\)\ \((\(-3\) + x\^2)\)\^4, {{{5, \(-5\)}, 
                    6}, "gS[3,{{4,8,3},{1,8,-1}}]"}}, 
{"3,16,48", \((\(-2\) + x)\)\^2\ \((\(-1\) + 
                      x)\)\ x\^4\ \((1 + x)\)\ \((2 + x)\)\^2\ \((3 + 
                      x)\)\ \((\(-3\) + x\^2)\)\^2, {{{\(-5\), 5, \(-3\), 
                      5, \(-7\), 7, 0, 3, \(-5\), 5}, 
                    1}, "gS[3,{{1,12},{.6,4}},{{0,7,5},2}]"}}, 
{"3,16,32,1", \((\(-1\) + x)\)\^2\ \((1 + 
                      x)\)\ \((\(-1\) - 2\ x + x\^2)\)\^2\ \((\(-1\) - 4\ x + 2\ x\^2 + 4\ x\^3 + x\^4)\)\^2, {{{8}, 
                    8}}}, 
{"3,16,32,2", \((\(-1\) + x)\)\^3\ \((1 + x)\)\^3\ \((3 + 
                      x)\)\ \((\(-1\) - 2\ x + x\^2)\)\^2\ \((\(-1\) + 2\ x + x\^2)\)\^2, {{{3, \(-3\)}, 
                    8}, "gP[8,1]"}}, 
{"3,16,24", \((\(-1\) + x)\)\^3\ \((1 + x)\)\^6\ \((\(-5\) + x\^2)\)\^3, {{{2, 
                      6, \(-6\), \(-2\)}, 
                    4}}}, 
{"3,16,16,1", \((\(-1\) + x)\)\^3\ \((1 + x)\)\^2\ \((\(-5\) + x\^2)\)\ \((\(-1\) - 2\ x + 
                      x\^2)\)\ \((\(-1\) + 2\ x + x\^2)\)\^3, {{{8, 
                      8, \(-7\), 7}, 
                    4}}}, 
{"3,16,16,2", \((\(-1\) + 
                      x)\)\ \((\(-5\) + 
                      x\^2)\)\ \((\(-1\) + 2\ x + x\^2)\)\^2\ \((1 - 4\ x\^2 + x\^4)\)\^2, {{{\(-3\), \(-6\), 7, \(-4\), 4, \(-7\), 6, 3}, 
                    1}, "gP[-4,1]"}}, 
{"3,16,8,1", \((\(-1\) + x)\)\ \((1 + x)\)\^2\ \((\(-3\) + x\^2)\)\ \((\(-1\) - 2\ x + 
                      
                      x\^2)\)\ \((\(-1\) - 8\ x - 4\ x\^2 + 2\ x\^3 + x\^4)\)\^2, {{{\(-2\), 8}, 
                    8}}}, 
{"3,16,8,2", \((\(-1\) + 
                      x)\)\ \((1 + x)\)\^2\ \((\(-3\) + x\^2)\)\ \((\(-1\) - 
                      2\ x + x\^2)\)\ \((3 - 4\ x - 4\ x\^2 + 2\ x\^3 + x\^4)\)\^2, {{{8, \(-6\), 8, 6}, 
                    2}}}, 
{"3,16,8,3", \((\(-1\) + x)\)\^3\ \((1 + x)\)\^3\ \((3 + x)\)\ \((\(-3\) + x\^2)\)\^2\ \((\(-1\) - 
                      2\ x + x\^2)\)\ \((\(-1\) + 2\ x + 
                      x\^2)\), {{{\(-5\), \(-3\)}, 
                    8}, "gS[3,{{1,16,1,-1.375}},{{-7,7,-7,-5,-7,7,-7,3},1}]"}}, 
{"3,16,8,4", \((\(-1\) + x)\)\^2\ \((1 + 
                      x)\)\ \((\(-5\) + x\^2)\)\ \((\(-3\) + 
                      x\^2)\)\ \((\(-1\) + 2\ x + 
                      x\^2)\)\ \((\(-1\) - 3\ x + x\^2 + x\^3)\)\^2, {{{\(-6\), 7, \(-7\), 6}, 
                    2}}}, 
{"3,16,8,5", \((\(-1\) + x)\)\^2\ \((1 + x)\)\^3\ \((\(-1\) - 2\ x + x\^2)\)\ \((\(-1\) + 2\ x + 
                      x\^2)\)\ \((\(-1\) - 5\ x + x\^2 + x\^3)\)\^2, {{{8, \(-3\), 3, 8}, 3}, {{\(-3\), \(-5\), 6, 6, \(-6\), \(-6\), 5, 3}, 
                    1}, {{\(-3\), 3, \(-3\), 8, 0, 3, 8, 3, 0, 3}, 
                    1}, {{\(-7\), 7, \(-3\), \(-5\), 6, 6, 6, 6}, 
                    1}, "gS[3,{{1.,8},{.5,8,4}},{{16,-1,-1}}]"}}, 
{"3,16,8,6", \((\(-1\) + x)\)\ \((1 + x)\)\ \((3 + 
                      x)\)\ \((\(-1\) - 2\ x + x\^2)\)\ \((\(-1\) + 2\ x + 
                      x\^2)\)\ \((1 - 4\ x\^2 + x\^4)\)\^2, {{{\(-7\), 
                      7, \(-3\), 3}, 
                    3}}}, 
{"3,16,8,7", \((\(-1\) + x)\)\^3\ \((1 + x)\)\^4\ \((\(-3\) - x + x\^2)\)\ \((\(-4\) + x + 
                      x\^2)\)\ \((4 - 9\ x - 6\ x\^2 + 2\ x\^3 + 
                      x\^4)\), {{{8, \(-2\), 8, 6, 8, 6, 8, 6}, 
                    1}}}, 
{"3,16,4,1", \((\(-1\) + x)\)\^2\ \((1 + x)\)\ \((\(-3\) + x\^2)\)\^2\ \((\(-1\) - 8\ x - 4\ x\^2 + 
                      2\ x\^3 + x\^4)\)\ \((3 - 4\ x - 4\ x\^2 + 2\ x\^3 + 
                      x\^4)\), {{{\(-4\), 8}, 6}, {{8, 4}, 
                    6}}}, 
{"3,16,4,2", \((\(-1\) + x)\)\^2\ \((1 + x)\)\ \((\(-1\) + 2\ x + x\^2)\)\^2\ \((5 - 8\ x\^2 + 
                      x\^4)\)\ \((1 - 4\ x\^2 + x\^4)\), {{{4, 4}, 
                    6}}}, 
{"3,16,4,3", \((\(-1\) + x)\)\^2\ \((1 + x)\)\ \((\(-1\) + 2\ x + x\^2)\)\^2\ \((13 - 8\ x\^2 + 
                      x\^4)\)\ \((1 - 4\ x\^2 + x\^4)\), {{{\(-4\), 4}, 
                    6}}}, 
{"3,16,4,4", \((1 - 2\ x - 
                      x\^2 + x\^3)\)\ \((\(-3\) + 3\ x\^2 + x\^3)\)\ \((1 + 
                      x - 5\ x\^2 + x\^4)\)\ \((1 + 12\ x - 4\ x\^2 - 
                      7\ x\^3 + x\^4 + x\^5)\), {{{\(-7\), 4, \(-5\)}, 
                    4}}}, 
{"3,16,4,5", \((\(-2\) + 
                      x)\)\ \((\(-1\) + x)\)\ x\^2\ \((1 + x)\)\ \((2 + 
                      x)\)\ \((\(-3\) + x\^2)\)\^2\ \((\(-2\) + 2\ x + 
                      x\^2)\)\ \((\(-2\) - 4\ x + x\^2 + 
                      x\^3)\), {{{8, \(-5\), 5, 8, 5, 8}, 
                    2}}}, 
{"3,16,4,6", \((3 + 
                      x)\)\ \((\(-2\) + x\^2)\)\^2\ \((\(-1\) - x + 
                      x\^2)\)\ \((\(-1\) + x + x\^2)\)\ \((\(-1\) - 4\ x + 
                      x\^3)\)\ \((1 - 4\ x + x\^3)\), {{{\(-7\), \(-5\), 5, 
                      7, 7, \(-7\)}, 
                    2}}}, 
{"3,16,2,1", \((\(-1\) + 
                      x)\)\ \((1 + x)\)\^2\ \((\(-3\) + x\^2)\)\ \((\(-1\) + 
                      2\ x + x\^2)\)\ \((1 - 20\ x - 36\ x\^2 + 24\ x\^3 + 
                      42\ x\^4 - 4\ x\^5 - 12\ x\^6 + 
                      x\^8)\), {{{\(-4\), \(-2\)}, 
                    8}}}, 
{"3,16,2,2", \((\(-1\) + 
                      x)\)\ \((1 + x)\)\^2\ \((\(-3\) + x\^2)\)\ \((\(-1\) + 
                      2\ x + x\^2)\)\ \((1 + 20\ x - 36\ x\^2 - 24\ x\^3 + 
                      42\ x\^4 + 4\ x\^5 - 12\ x\^6 + 
                      x\^8)\), {{{\(-4\), \(-6\), 4, 6}, 
                    3}}}, 
{"3,16,1", \((1 + 
                      x)\)\ \((\(-3\) - 16\ x + 110\ x\^2 + 20\ x\^3 - 
                      509\ x\^4 + 206\ x\^5 + 668\ x\^6 - 292\ x\^7 - 
                      389\ x\^8 + 140\ x\^9 + 115\ x\^10 - 28\ x\^11 - 
                      17\ x\^12 + 2\ x\^13 + x\^14)\), {{{5, 5, 11, 5, 5, 0, 
                      0, 4, 0, 0, 4, 0, 3}, 
                    1}}}, 
{"P316", 
                x\^3\ \((1 + x)\)\^4\ \((\(-4\) + x + 
                      x\^2)\)\ \((2 - 6\ x - x\^2 + x\^3)\)\^2, {"gS[3,{{2,12,1,1.32},{1,3,1,0},{0,1}},{{-1,2,4,4,3},{-13,14},{-13,15},{-14,15},{2,4,4,4,3},{3,5,4,4,2},{2,13,4,1,3},{5,13,4,1,3},{1,11,0,4,2}}]"}}\)},{},
{  \({"Pappus 216", 
                x\^4\ \((3 + x)\)\ \((\(-3\) + x\^2)\)\^6, {{{5, 7, \(-7\), 
                      7, \(-7\), \(-5\)}, 
                    3}, "gS[3,{{.82,6},{1,12,1,-.262,-1}},{{-7,8,4,4,3},{2,8,2,4,3},{1,7,2,4,3},{8,5,-3},{9,5,1}}]"}}, 
{"3,18,72
ct 23", \((\(-1\) + 
                      x)\)\ x\^4\ \((2 + x)\)\^4\ \((\(-3\) - x + x\^2)\)\^4, {"gS[3,{{2.,12,1,Pi/12.},{1.,6,3}}]", "gS[3,{{1.,18}},{{-9,-2,2},5}]"}}, 
{"3,18,48", \((\(-1\) + x)\)\^5\ x\^4\ \((2 + x)\)\^4\ \((\(-6\) + x\^2)\)\^2, {{{\(-8\), 8, 
                      9}, 3}}}, 
{"3,18,36", \((\(-1\) + x)\)\ x\^2\ \((2 + x)\)\^2\ \((3 - 3\ x\^2 + x\^3)\)\^2\ \((\(-1\) + 3\ x\^2 + x\^3)\)\^2, {{{\(-8\), 8, \(-3\), \(-5\), \(-7\), 9, 7, 5, 3}, 1, 
                    4}}}, 
{"3,18,18
ct 20", \((3 + 
                      x)\)\ \((\(-3\) + x\^2)\)\^2\ \((\(-3\) + 9\ x\^2 - 6\ x\^4 + x\^6)\)\^2, {{{\(-7\), 7}, 
                    5}}}, 
{"3,18,16", \((\(-1\) + x)\)\^2\ \((1 + x)\)\^2\ \((3 + x)\)\ \((\(-1\) - 2\ x + x\^2)\)\ \((\(-3\) - x + 
                      x\^2)\)\ \((\(-1\) - x + x\^2)\)\ \((\(-3\) + x + 
                      x\^2)\)\ \((\(-1\) + x + x\^2)\)\ \((\(-1\) + 2\ x + 
                      x\^2)\), {{{9, \(-5\), \(-5\), 9, 9, \(-5\), 5}, 
                    6}}}, 
{"3,18,12,1", 
                x\^4\ \((3 + 
                      x)\)\ \((\(-3\) + x\^2)\)\^2\ \((3 - 6\ x\^2 + x\^4)\)\^2, {{{3, 9, 3}, 5}, {{\(-9\), \(-3\), 9}, 
                    5}}}, 
{"3,18,12,2", \((\(-2\) + 
                      x)\)\ \((\(-1\) + x)\)\^2\ x\^2\ \((1 + x)\)\^2\ \((2 + 
                      x)\)\ \((3 + 
                      x)\)\ \((3 - 6\ x\^2 + x\^4)\)\^2, {{{\(-7\), 9, 7}, 
                    3}, {{5, 3, 7, \(-7\), \(-3\), \(-5\)}, 
                    3}, "gP[-6,-6]"}}, 
{"3,18,12,3", \((1 + x)\)\^3\ \((\(-3\) + 
                      x\^2)\)\ \((\(-1\) - x + x\^2)\)\^2\ \((\(-1\) + 2\ x + 
                      x\^2)\)\ \((3 - 6\ x + x\^3)\)\^2, {"gS[3,{{1.5,12,3},{1.,6,1}},{{2,2,2}}]"}}, 
{"3,18,12,4", \((\(-1\) + x)\)\^3\ x\^2\ \((1 + x)\)\^2\ \((2 + x)\)\^2\ \((\(-7\) + 
                      x\^2)\)\ \((\(-3\) + x\^2)\)\^3, {"gS[3,{{1.,3,1,.525},{2.3,12,2,.52},{1.,3,1,2.1}},{{6,5,-1}}]"}}, 
{"3,18,8,1", \((1 + x)\)\ \((\(-3\) + 
                      x\^2)\)\ \((\(-1\) - x + x\^2)\)\ \((1 + 3\ x + 
                      x\^2)\)\ \((\(-3\) + 4\ x + 7\ x\^2 - 5\ x\^3 - 
                      2\ x\^4 + x\^5)\)\ \((1 + 6\ x - 7\ x\^2 - 5\ x\^3 + 
                      2\ x\^4 + 
                      x\^5)\), {"gS[3,{{1.,8},{.6,6},{.3,4,1,3.π/4}},{{2,2,2},{3,4,1}}]"}}, 
{"3,18,6,1", \((1 + x)\)\^3\ \((\(-3\) + 
                      x\^2)\)\ \((\(-1\) - 2\ x + 
                      x\^2)\)\ \((3 + 9\ x - 6\ x\^2 - 7\ x\^3 + x\^4 + x\^5)\)\^2, {"gS[3,{{1,3,1,1. π},{2.7,9,1,-1.745},{1.9,3,0},{1,3}}]"}}, 
{"3,18,6,2", \((\(-1\) + x)\)\^3\ \((\(-3\) - x + 
                      x\^2)\)\ \((1 + 3\ x + 
                      x\^2)\)\ \((3 - 10\ x\^2 - 5\ x\^3 + 2\ x\^4 + x\^5)\)\^2, {"gS[3,{{3.,6,1},{2.,9,1,.52},{1.,3,1,-1.52}}]"}}, 
{"3,18,6,3", \((\(-2\) + 
                      x)\)\ x\^3\ \((1 + x)\)\^2\ \((\(-4\) - 6\ x + x\^2 + 
                      x\^3)\)\ \((6 - 4\ x - 6\ x\^2 + x\^3 + x\^4)\)\^2, {"gS[3,{{1.,3,1,.525},{2,12,3,.52},{1.,3,1,2.1}},{{6,5,-1}}]"}}, 
{"3,18,6,4", \((\(-2\) + x)\)\ x\^2\ \((2 + 
                      x)\)\ \((3 + 
                      x)\)\ \((\(-3\) + 12\ x\^2 - 7\ x\^4 + x\^6)\)\^2, {"gS[3,{{.52,6},{1,12,1,-.262,-1}},{{-7,8,4,4,3},{2,8,2,4,3},{1,7,2,4,3},{9,5,-1},{7,5,-1}}]"}}, 
{"3,18,4,1", \((\(-2\) + 
                      x)\)\ x\^2\ \((2 + x)\)\ \((3 + x)\)\ \((3 - 3\ x\^2 + 
                      x\^3)\)\ \((1 - 2\ x - x\^2 + x\^3)\)\ \((\(-1\) - 
                      2\ x + x\^2 + x\^3)\)\ \((\(-3\) + 3\ x\^2 + 
                      x\^3)\), {{{9, \(-5\), 9, 9, 9, 5}, 
                    3, \(-3\)}}}, 
{"3,18,4,2", \((\(-1\) + x)\)\ x\^4\ \((1 + x)\)\ \((3 + x)\)\ \((\(-3\) + x\^2)\)\^2\ \((3 - 
                      5\ x - x\^2 + x\^3)\)\ \((\(-3\) - 5\ x + x\^2 + 
                      x\^3)\), {{{9, \(-3\), 9, 3}, 
                    
                    9, \(-2\)}}}, 
{"3,18,4,3", \((\(-1\) + x)\)\ x\^2\ \((1 + x)\)\ \((\(-3\) + x\^2)\)\ \((\(-4\) + x + 
                      x\^2)\)\ \((\(-1\) + 2\ x + x\^2)\)\ \((3 - 5\ x - 
                      x\^2 + x\^3)\)\ \((4 - 3\ x - 5\ x\^2 + x\^3 + 
                      x\^4)\), {{{\(-6\), 9, 9, 6}, 5, 
                    1}}}, 
{"3,18,4,4", 
                x\^4\ \((1 + x)\)\ \((\(-3\) + x\^2)\)\ \((\(-1\) - 2\ x + 
                      x\^2)\)\ \((9 - 8\ x - 6\ x\^2 + 2\ x\^3 + x\^4)\)\^2, {"gS[3,{{1.,18}},{{1,6,1},{2,3,4},{13,3,0}}]"}}, 
{"3,18,4,5", \((\(-1\) + x)\)\ \((1 + 
                      x)\)\ \((\(-3\) + x\^2)\)\ \((\(-1\) - x + 2\ x\^2 + 
                      x\^3)\)\ \((6 + 5\ x - 7\ x\^2 - x\^3 + 
                      x\^4)\)\ \((\(-2\) + 7\ x + 9\ x\^2 - 9\ x\^3 - 
                      6\ x\^4 + 2\ x\^5 + 
                      x\^6)\), {"gS[3,{{1,9},{.45,6},{.2,3,1,3.14}},{{2,5,4,4,2},{3,5,0}}]"}}, 
{"3,18,2,1", \((\(-1\) + x)\)\ x\ \((5 + 2\ x - 
                      16\ x\^2 - 12\ x\^3 + 4\ x\^4 + 5\ x\^5 + 
                      x\^6)\)\ \((12 + 53\ x + 8\ x\^2 - 95\ x\^3 - 
                      22\ x\^4 + 56\ x\^5 + 9\ x\^6 - 13\ x\^7 - x\^8 + 
                      x\^9)\), {"gS[3,{{1.,5,1,.7π},{.7,5,1,.7π},{.4,5,1,2.2},{.11,3,1,2.6}},{{3,2,2}}]"}}, 
{"3,18,2,2", \((\(-1\) + x)\)\ \((3 + 
                      2\ x - 40\ x\^2 - 12\ x\^3 + 48\ x\^4 + 2\ x\^5 - 
                      13\ x\^6 + x\^8)\)\ \((\(-1\) - 2\ x + 8\ x\^2 + 
                      12\ x\^3 - 12\ x\^4 - 18\ x\^5 - x\^6 + 4\ x\^7 + 
                      x\^8)\), {{{9, \(-6\), \(-8\), 8, 6}, 4, 
                    2}}}, 
{"3,18,2,3", \((\(-1\) + x)\)\^2\ x\ \((2 + x)\)\ \((\(-6\) + 8\ x + 17\ x\^2 - 6\ x\^3 - 8\ x\^4 + 
                      x\^5 + x\^6)\)\ \((\(-6\) - 12\ x + 33\ x\^2 + 
                      23\ x\^3 - 16\ x\^4 - 9\ x\^5 + 2\ x\^6 + 
                      x\^7)\), {{{9, \(-5\), 8, 9, \(-5\), 8, 9, 4, 8}, 
                    1}}}, 
{"3,18,1", \((\(-1\) + 
                      x)\)\ x\ \((2 + x)\)\ \((\(-3\) + x\^2)\)\ \((\(-16\) - 
                      92\ x - 12\ x\^2 + 300\ x\^3 + 128\ x\^4 - 328\ x\^5 - 
                      160\ x\^6 + 152\ x\^7 + 76\ x\^8 - 30\ x\^9 - 
                      15\ x\^10 + 2\ x\^11 + 
                      x\^12)\), {{{\(-2\), \(-5\), \(-3\), 8, \(-7\), 5, 7, 
                      2, 4}, 
                    1}}}, 
{"3,18,nx ", \((\(-3\) + x)\)\^2\ \((\(-1\) + x)\)\^3\ x\^6\ \((2 + x)\)\^6, {"gS[3,{{1.,3,1,.525},{2.3,12,4,.52},{1.,3,1,2.1}},{{6,5,-1}}]"}}\)},{},
{  \({"3,20,320", \((\(-1\) + x)\)\^5\ \((1 + x)\)\^5\ \((3 + x)\)\ \((11 - 8\ x\^2 + x\^4)\)\^2, {{{11, 9}, 5}}}\),
\({"Desargues 240", \((\(-2\) + x)\)\^4\ \((\(-1\) + x)\)\^5\ \((1 + x)\)\^5\ \((2 + x)\)\^4\ \((3 + 
                        x)\), {{{5, \(-5\), 9, \(-9\)}, 
                      10}, "gS[3,{{1.,10},{.6,10,3}}]"}}\),
\({"F20 12
Dodecahedral", \((\(-1\) + x)\)\^5\ x\^4\ \((2 + x)\)\^4\ \((\(-5\) + x\^2)\)\^3, {"gS[3,{{1.,5,1,-π/2.},{.6,10,1,-π/2.},{.3,5,1,π/2.}}]", {{\(-4\), \(-7\), 10, \(-4\), 7, \(-7\), 4, 10, 7, 4}, 1}, "gP[10,2]"}}\),
\({"3,20,20,1", \((\(-1\) + 
                        x)\)\ \((1 + x)\)\ \((3 + 
                        x)\)\ \((\(-1\) - x + x\^2)\)\^2\ \((\(-1\) + x + x\^2)\)\^2\ \((11 - 7\ x\^2 + x\^4)\)\^2, {{{13, 7}, 6}}}\),
\({"3,20,20,2", \((1 + x)\)\^3\ \((2 + 2\ x - 6\ x\^2 + x\^4)\)\^4, {{{14, 0, 0, 6}, 5}}}\),
(*Largest cubic non - planar graph with dia = 
                      3 (see MathWorld LCF)*)
\({"3,20,20,3", \((\(-2\) + 
                        x)\)\ \((1 + x)\)\ \((2 + 
                        x)\)\ \((\(-1\) - x + x\^2)\)\^4\ \((\(-1\) - 11\ x - 3\ x\^2 + 3\ x\^3 + x\^4)\)\^2, {{\(-6\), 7, 10, \(-7\), \(-5\), 4, 7, 
                      10, \(-7\), \(-4\), 5, 7}, 1}}\),
\({"3,20,20,4", \((\(-1\) + 
                        x)\)\ \((\(-5\) + 
                        x\^2)\)\ \((\(-1\) - x + x\^2)\)\^2\ \((1 + 3\ x + x\^2)\)\^2\ \((5 - 5\ x\^2 + x\^4)\)\^2, {{{\(-6\), 5, \(-6\), \(-4\), 9, 5, 
                        0, \(-9\), 4, 6, 0, 6}, 1}}}\),\({"3,20,20,5", \((\(-2\) + x)\)\^2\ \((1 + 
                        x)\)\ \((\(-1\) - x + x\^2)\)\^2\ \((\(-1\) + x + x\^2)\)\^2\ \((\(-1\) - 11\ x - 3\ x\^2 + 3\ x\^3 + x\^4)\)\^2, {"gS[3,{{.25,5,3,-.22},{1.,5,0},{.55,5,2,-.15},{.74,-5,1,-.3}}]"}}\),
\({"FlowerSnark 20", \((\(-2\) + 
                        x)\)\ \((1 + x)\)\ \((2 + 
                        x)\)\ \((\(-1\) - x + x\^2)\)\^2\ \((\(-1\) + x + x\^2)\)\^2\ \((9 - 3\ x - 7\ x\^2 + x\^3 + x\^4)\)\^2, {"gS[3,{{3,15},{1,5}},{{3,5,1}}]"}}\),
\({"3,20,10", \((\(-1\) + 
                        x)\)\ \((1 + x)\)\ \((3 + 
                        x)\)\ \((1 - 15\ x\^2 + 28\ x\^4 - 10\ x\^6 + x\^8)\)\^2, {{{\(-9\), \(-3\), 3, 9}, 8}, {{7, 5, 0, 0, 7, 5}, 5}}}\)},{  },
{  
\({"3,22,22,1", \((\(-1\) + 
                      x)\)\ \((23 + 74\ x + 17\ x\^2 - 132\ x\^3 - 73\ x\^4 + 84\ x\^5 + 49\ x\^6 - 22\ x\^7 - 12\ x\^8 + 2\ x\^9 + x\^10)\)\^2, {"gP[11,3]", {{\(-6\), 10, \(-5\), 6, 9, \(-7\), 11, 7, \(-9\), 0, 5, 0, 
                      6}, 1, \(-6\)}}}, 
{"3,22,22 nh", \((\(-1\) + 
                      x)\)\ \((1 - 14\ x + 61\ x\^2 - 66\ x\^3 - 84\ x\^4 + 73\ x\^5 + 49\ x\^6 - 22\ x\^7 - 12\ x\^8 + 2\ x\^9 + x\^10)\)\^2, {"gP[11,5]", "gP[11,2]", "gS[3,{{1,21,1,.22},{0,1,0}},{{21,10,17,14,11,8,15,13,10,7,4,0,9},1}]"}}, 
{"3,22,22,3", 
                12 + 72\ x - 497\ x\^2 - 949\ x\^3 + 4063\ x\^4 + 
                  4119\ x\^5 - 13473\ x\^6 - 8847\ x\^7 + 22669\ x\^8 + 
                  11013\ x\^9 - 21271\ x\^10 - 8565\ x\^11 + 11664\ x\^12 + 
                  4232\ x\^13 - 3788\ x\^14 - 1304\ x\^15 + 714\ x\^16 + 
                  240\ x\^17 - 72\ x\^18 - 24\ x\^19 + 3\ x\^20 + 
                  x\^21, {{{\(-7\), \(-3\), \(-8\), 10, 8, \(-8\), 5, 11, 9, 
                      5, 11}, 1}}}\)},{},
{  \({"F24 144", \((\(-2\) + x)\)\^6\ \((\(-1\) + x)\)\^3\ x\^4\ \((1 + x)\)\^3\ \((2 + x)\)\^6\ \((3 + 
                      x)\), {"gS[3,{{1.,12},{.6,12,5}}]", {{5, \(-9\), 
                      7, \(-7\), 9, \(-5\)}, 
                    4}}}, 
{"3,24,48,1", \((\(-2\) + x)\)\^2\ \((\(-1\) + x)\)\^2\ x\^2\ \((1 + 
                      x)\)\ \((\(-2\) - 2\ x + x\^2)\)\^2\ \((\(-1\) + 2\ x + x\^2)\)\^2\ \((\(-2\) - 4\ x + 2\ x\^2 + 4\ x\^3 + x\^4)\)\^2, {{{12}, 
                    12}}}, 
{"3,24,48,2", \((\(-1\) + 
                      x)\)\ \((\(-5\) + 
                      x\^2)\)\ \((\(-3\) - x + x\^2)\)\^2\ \((\(-1\) - x + x\^2)\)\^4\ \((\(-1\) + 2\ x + x\^2)\)\^2\ \((1 + 3\ x + x\^2)\)\^2, {"gP[-6,-3]", {{\(-7\), 8, 12, 12, \(-8\), 7, \(-7\), \(-9\), 8, 0, 9, 7, 
                      0, 8}, 1}}}, 
{"Truncated-
Octahedral 48", \((\(-1\) + x)\)\^3\ \((1 + x)\)\^3\ \((3 + 
                      x)\)\ \((\(-3\) + x\^2)\)\^2\ \((\(-1\) - 2\ x + x\^2)\)\^3\ \((\(-1\) + 2\ x + x\^2)\)\^3, {"gS[3,{{5,4,1,.78},{4,4,0,.785},{2.8,8,1,-.39},
{2,4,0,- .78},{1,4,1,-.78}},{{-9,10,2,2,4},{6,9,1,2,4},{6,10,1,2,4},{5,10,0},{5,11,0}}]"}}, 
{"McGee 32
Cage(3,7)", \(\((\(-2\) + x)\)\^3\) 
                  x\^3\ \((1 + x)\)\^2\ \((2 + x)\)\ \((\(-4\) + x + 
                      x\^2)\)\ \((\(-2\) - 4\ x + x\^2 + x\^3)\)\^4, {{{12, 
                      7, \(-7\)}, 
                    6\ }}}, 
{"3,24,24,1", \((\(-2\) + x)\)\^3\ x\^6\ \((1 + x)\)\^3\ \((2 + x)\)\^7\ \((\(-5\) - x + x\^2)\)\^2\ \((\(-3\) - x + x\^2)\)\^2\ \((1 + 4\ x - 3\ x\^2 - 2\ x\^3 + x\^4)\)\^2, {"gP[12,6]"}}, 
{"3,24,24,2", \((\(-1\) + 
                      x)\)\ x\^2\ \((2 + x)\)\^2\ \((\(-5\) + 
                      x\^2)\)\ \((\(-2\) + x\^2)\)\^2\ \((\(-1\) + 2\ x + x\^2)\)\^2\ \((\(-2\) + 8\ x - 4\ x\^2 - 2\ x\^3 + x\^4)\)\^2, {{{\(-5\), \(-8\), \(-11\), \(-4\), 10, \(-7\), 7, \(-10\), 4, 11, 8, 5}, 
                    1}, "gP[-6,1]"}}, 
{"3,24,24,3", \((\(-1\) + x)\)\^3\ \((1 + x)\)\^3\ \((3 + x)\)\ \((\(-3\) - x + x\^2)\)\^2\ \((\(-3\) + x + x\^2)\)\^2\ \((1 - 5\ x\^2 + x\^4)\)\^2, {"gS[3,{{1,24}},{{13,-13,5,-5},6}]"}}, 
{"3,24,16,1", \((\(-1\) + x)\)\^2\ \((1 + 
                      x)\)\ \((\(-7\) + x\^2)\)\ \((\(-3\) + 
                      x\^2)\)\ \((\(-1\) + 2\ x + x\^2)\)\^2\ \((1 - 3\ x - x\^2 + x\^3)\)\^2\ \((\(-1\) - 3\ x + x\^2 + x\^3)\)\^2, {"gS[3,{{.4,4,1},{.8,8},{.6,8,1,2.65},{.4,4,1,-.5}}]"}}, 
{"3,24,16,2", \((\(-2\) + x)\)\^2\ \((\(-1\) + 
                      x)\)\ x\^4\ \((1 + x)\)\ \((2 + x)\)\^2\ \((3 + 
                      x)\)\ \((\(-2\) + x\^2)\)\^2\ \((2 - 7\ x\^2 + x\^4)\)\^2, {"gP[-8,-8]"}}, 
{"3,24,6", 
                x\^2\ \((\(-7\) - 6\ x + x\^2 + x\^3)\)\^2\ \((\(-1\) + 
                      3\ x + 4\ x\^2 + 
                      x\^3)\)\ \((2 - 4\ x\^2 + x\^4)\)\^2\ \((\(-1\) + 
                      6\ x - x\^2 - 3\ x\^3 + 
                      x\^4)\), {"gS[3,{{1.5,6},{1.,12,1,-.26},{.6,3,0,2.63},{.25,3,1,2.62}}]"}}, 
{"Truncated-
Cubical 48", \((\(-2\) + x)\)\^3\ \((\(-1\) + 
                      x)\)\ x\^5\ \((1 + x)\)\^3\ \((2 + x)\)\^5\ \((\(-4\) - x + x\^2)\)\^3, {"gS[3,{{8,8,1,.393},{6,4,0},{4,4,0},{2,8,1,.393}}]"}}\)},{},
{  \({"F26 78", \((3 + 
                      x)\)\ \((3 - 5\ x\^2 + x\^4)\)\^6, {{{7, \(-7\)}, 
                    13}, "gS[3,{{1,18},{1.5,6},{.4,-2,-1,-.32,-2}},{{2,25,6,0,3},{5,26,6,0,3},{3,12,3,3,3}}]"}}, 
{"3,26,52", \((3 + x)\)\ \((1 - 2\ x - 7\ x\^2 + 6\ x\^3 + 5\ x\^4 - 5\ x\^5 + x\^6)\)\^2\ \((1 + 2\ x - 7\ x\^2 - 6\ x\^3 + 5\ x\^4 + 5\ x\^5 + x\^6)\)\^2, {{{13}, 
                    13}}}, 
{"3,26,26,1", \((3 + 
                      
                      x)\)\ \((53 - 213\ x\^2 + 317\ x\^4 - 228\ x\^6 + 84\ x\^8 - 15\ x\^10 + x\^12)\)\^2, {{{9, \(-9\)}, 13}, {{11, \(-11\)}, 
                    8}}}, 
{"3,26,26,2", \((\(-1\) + 
                      x)\)\ \((1 + 31\ x - 149\ x\^2 + 102\ x\^3 + 270\ x\^4 - 231\ x\^5 - 210\ x\^6 + 131\ x\^7 + 82\ x\^8 - 28\ x\^9 - 15\ x\^10 + 2\ x\^11 + x\^12)\)\^2, {"gP[13,2]", "gP[13,6]"}}, 
{"3,26,26,3", \((\(-1\) + 
                      x)\)\ \((\(-25\) - 125\ x - 32\ x\^2 + 349\ x\^3 + 192\ x\^4 - 335\ x\^5 - 197\ x\^6 + 144\ x\^7 + 82\ x\^8 - 28\ x\^9 - 15\ x\^10 + 2\ x\^11 + x\^12)\)\^2, {"gP[13,3]", "gP[13,4]"}}, 
{"3,26,26,4", \((\(-1\) + 
                      x)\)\ \((1 + 9\ x + 15\ x\^2 - 6\ x\^3 - 8\ x\^4 + x\^5 + x\^6)\)\^4, {"gP[13,5]"}}, 
{"3,26,26,5", \((\(-1\) + x)\)\^3\ \((84 - 1064\ x - 3324\ x\^2 + 10232\ x\^3 + 28520\ x\^4 - 
                      27860\ x\^5 - 91320\ x\^6 + 20760\ x\^7 + 
                      136844\ x\^8 + 14684\ x\^9 - 110480\ x\^10 - 
                      33288\ x\^11 + 50764\ x\^12 + 22816\ x\^13 - 
                      13032\ x\^14 - 8020\ x\^15 + 1600\ x\^16 + 
                      1540\ x\^17 - 16\ x\^18 - 152\ x\^19 - 15\ x\^20 + 
                      6\ x\^21 + x\^22)\), {{{11, \(-12\), 12, \(-11\)}, 
                    10}}}, 
{"3,26,2,1", \((7 - 2\ x - 
                      94\ x\^2 - 14\ x\^3 + 248\ x\^4 + 34\ x\^5 - 
                      239\ x\^6 - 18\ x\^7 + 99\ x\^8 + 2\ x\^9 - 17\ x\^10 + 
                      x\^12)\)\ \((1 - 3\ x - 28\ x\^2 + 40\ x\^3 + 
                      214\ x\^4 - 10\ x\^5 - 339\ x\^6 - 69\ x\^7 + 
                      187\ x\^8 + 53\ x\^9 - 41\ x\^10 - 13\ x\^11 + 
                      3\ x\^12 + 
                      x\^13)\), {"gS[3,{{1.,26,1,π/13.}},{{1,12,-3},{9,9,-1}}]"}}, 
{"3,26,2,2", \((3 + x)\)\ \((\(-1\) - x + 
                      x\^2)\)\ \((\(-1\) + x + x\^2)\)\ \((\(-4\) - 8\ x + 
                      49\ x\^2 + 33\ x\^3 - 94\ x\^4 - 31\ x\^5 + 57\ x\^6 + 
                      10\ x\^7 - 13\ x\^8 - x\^9 + x\^10)\)\ \((\(-4\) + 
                      8\ x + 49\ x\^2 - 33\ x\^3 - 94\ x\^4 + 31\ x\^5 + 
                      57\ x\^6 - 10\ x\^7 - 13\ x\^8 + x\^9 + 
                      x\^10)\), {"gS[3,{{1.,26}},{{1,11,-2},{9,9,-1}}]"}}\)},{},
{  \({"Coxeter 336", \((\(-2\) + x)\)\^8\ \((1 + x)\)\^7\ \((\(-1\) + 2\ x + x\^2)\)\^6, {"gS[3,{{.2,7,3,.175},{1.,7,0},{.5,7,2,.35},{.7,-7,1,-.44}},{}]", "gS[3,{{.65,21,1,Pi/14.},{1.,7,1,Pi/14.}},{{-2,3,3,3,7},{2,8,3,3,5},{17,2,3,3,2},{3,12,3,3,7},{3,15,3,3,7}}]"}}, 
{"3,28,28,1", \((\(-1\) + x)\)\ \((\(-5\) + 
                      x\^2)\)\ \((1 + 7\ x + 12\ x\^2 - 8\ x\^4 + x\^6)\)\^2\ \((\(-1\) + 11\ x + 10\ x\^2 - 10\ x\^3 - 6\ x\^4 + 2\ x\^5 + x\^6)\)\^2, {"gP[-7,-7]"}}, 
{"3,28,28,2", \((\(-1\) + 
                      x)\)\ \((1 + x)\)\ \((3 + 
                      x)\)\ \((\(-2\) + x\^2)\)\^6\ \((\(-8\) + 24\ x\^2 - 10\ x\^4 + x\^6)\)\^2, {{{11, 13, \(-13\), 11}, 
                    7}}}, 
{"3,28,14", \((\(-1\) + 
                      x)\)\ \((1 + x)\)\ \((3 + 
                      x)\)\ \((1 - x - 2\ x\^2 + x\^3)\)\^2\ \((\(-1\) - x + 2\ x\^2 + x\^3)\)\^2\ \((\(-29\) + 31\ x\^2 - 10\ x\^4 + x\^6)\)\^2, {{{11, \(-7\), 7, \(-11\)}, 7}}}, 
{"3,28,12", 
                x\^3\ \((1 + x)\)\^3\ \((2 + x)\)\^4\ \((\(-4\) - x + x\^2)\)\^2\ \((\(-1\) - x + x\^2)\)\ \((4 - 5\ x - x\^2 + 
                      x\^3)\)\ \((2 + 6\ x - 4\ x\^2 - 2\ x\^3 + x\^4)\)\^2, {"gS[3,{{1.,18},{.6,9,0,-1.2,3},{0,1}},{{20,4,-1},{21,5,0}}]"}}, 
{"3,28,6", \((1 + x)\)\ \((\(-1\) - x + 2\ x\^2 + 
                      x\^3)\)\ \((\(-1\) + 13\ x - x\^2 - 8\ x\^3 + 
                      x\^5)\)\ \((1 + 19\ x - 11\ x\^2 - 70\ x\^3 + 5\ x\^4 + 52\ x\^5 - 13\ x\^7 + x\^9)\)\^2, {"gS[3,{{1.,24,1},{.5,3,0,-.1},{.0,1,-1,0}},{{3,5,3},{4,10,-2},{5,10,-2},{26,-3,0},{27,-20,0},{26,-13,0},{27,-6,0}}]"}}, 
{"3,28,2", \((1 + x)\)\ \((1 + 28\ x - 57\ x\^2 - 
                      104\ x\^3 + 203\ x\^4 + 106\ x\^5 - 212\ x\^6 - 
                      38\ x\^7 + 90\ x\^8 + 4\ x\^9 - 16\ x\^10 + 
                      x\^12)\)\ \((1 + 50\ x + 4\ x\^2 - 402\ x\^3 - 
                      222\ x\^4 + 880\ x\^5 + 575\ x\^6 - 700\ x\^7 - 
                      426\ x\^8 + 238\ x\^9 + 134\ x\^10 - 36\ x\^11 - 
                      19\ x\^12 + 2\ x\^13 + x\^14)\), {{{10, \(-5\), 
                      5, \(-10\)}, 7}}}\)},{},
{  \({"Levi F30 1440", \((\(-2\) + x)\)\^9\ x\^10\ \((2 + x)\)\^9\ \((3 + x)\), {{{7, \(-7\), 9, 13, \(-13\), \(-9\)}, 
                    5}}}, 
{"3,30,60", \((\(-1\) + 
                      x)\)\ x\^2\ \((2 + x)\)\^2\ \((\(-5\) + x\^2)\)\^4\ \((\(-1\) - x + x\^2)\)\^6\ \((1 + 3\ x + x\^2)\)\^2, {"gP[15,11]"}}, 
{"3,30,30,1", \((\(-1\) + x)\)\^5\ x\^2\ \((2 + x)\)\^6\ \((\(-5\) + 25\ x - 5\ x\^2 - 65\ x\^3 + 31\ x\^4 + 22\ x\^5 - 11\ x\^6 - 2\ x\^7 + x\^8)\)\^2, {"gP[15,2]", "gP[15,7]"}}, {"3,30,12", 
                x\^4\ \((3 + x)\)\ \((1 - 3\ x + x\^2)\)\ \((\(-1\) - x + 
                      x\^2)\)\ \((\(-1\) + x + x\^2)\)\ \((1 + 3\ x + 
                      x\^2)\)\ \((5 + 2\ x - 6\ x\^2 - x\^3 + x\^4)\)\^2\ \((5 - 2\ x - 6\ x\^2 + x\^3 + x\^4)\)\^2, {"gS[3,{{.1,30}},{{19,17,15,13,11},3}]"}}, 
{"3,30,6", \((\(-1\) + 
                      x)\)\ \((1 + x)\)\^2\ \((\(-1\) - x + 
                      x\^2)\)\ \((\(-3\) + x + x\^2)\)\ \((\(-1\) + x + 
                      x\^2)\)\ \((1 + 3\ x + 
                      x\^2)\)\ \((\(-18\) + 27\ x + 98\ x\^2 - 101\ x\^3 - 64\ x\^4 + 64\ x\^5 + 14\ x\^6 - 14\ x\^7 - x\^8 + x\^9)\)\^2, {{{\(-6\), 5, 
                      15, \(-5\), 6}, 
                    6}}}, 
{"3,30,,3", 
                x\^5\ \((2 + x)\)\^6\ \((\(-4\) - x + x\^2)\)\^5\ \((\(-1\) - x + x\^2)\)\^4, {"gS[3,{{1,18},{.8,9,0,-1.2,3},{.5,3,1,-1.2}},{{20,4,-1},{21,5,0}}]"}}\)},{},
{  \({"Dyck F32 192", \((\(-1\) + x)\)\^9\ \((1 + x)\)\^9\ \((3 + x)\)\ \((\(-5\) + x\^2)\)\^6, {{{13, \(-13\), 5, \(-5\)}, 
                    8}}}, 
{"3,32,64,1", \((\(-1\) + 
                      x)\)\ \((\(-5\) + 
                      x\^2)\)\ \((\(-1\) + 2\ x + x\^2)\)\^2\ \((1 - 4\ x\^2 + x\^4)\)\^2\ \((1 - 24\ x - 52\ x\^2 + 8\ x\^3 + 44\ x\^4 - 12\ x\^6 + x\^8)\)\^2, {{{16}, 16}, 
{{\(-6\), 8, \(-11\), 11, \(-8\), 
                      6, \(-13\), 14, 7, 0, 8, 0, \(-13\), 7, 0, 0, 14, 7, 0, 
                      8, 0, 0, 7}, 
                    1}}}, 
{"3,32,64,2", \((\(-1\) + x)\)\^3\ \((1 + x)\)\^3\ \((3 + 
                      x)\)\ \((\(-1\) - 2\ x + x\^2)\)\^2\ \((\(-1\) + 2\ x + x\^2)\)\^2\ \((\(-1\) + 4\ x + 2\ x\^2 - 4\ x\^3 + x\^4)\)\^2\ \((\(-1\) - 4\ x + 2\ x\^2 + 4\ x\^3 + x\^4)\)\^2, {{{\(-3\), 3}, 
                    16}}}, 
{"3,32,32,1", \((\(-1\) + x)\)\^3\ \((1 + x)\)\^3\ \((3 + 
                      x)\)\ \((\(-3\) + x\^2)\)\^4\ \((1 - 36\ x\^2 + 42\ x\^4 - 12\ x\^6 + x\^8)\)\^2, {"gP[16,3]"}}, {"3,32,32,2", \((\(-1\) + x)\)\ \((\(-5\) + 
                      x\^2)\)\ \((\(-1\) + 2\ x + x\^2)\)\^2\ \((1 - 4\ x\^2 + x\^4)\)\^2\ \((1 + 24\ x - 52\ x\^2 - 8\ x\^3 + 44\ x\^4 - 12\ x\^6 + x\^8)\)\^2, {"gP[-8,1]"}}, 
{"3,32,32,3", \((\(-1\) + x)\)\^2\ \((1 + 
                      x)\)\ \((\(-1\) - 2\ x + x\^2)\)\^2\ \((7 - 6\ x\^2 + x\^4)\)\^2\ \((\(-11\) - 14\ x + 44\ x\^2 + 56\ x\^3 - 12\ x\^4 - 30\ x\^5 - 
                      4\ x\^6 + 4\ x\^7 + x\^8)\)\ \((1 - 26\ x + 4\ x\^2 + 
                      48\ x\^3 - 26\ x\^5 - 4\ x\^6 + 4\ x\^7 + 
                      x\^8)\), {{{\(-8\), 16}, 
                    16}}}, 
{"3,32,16", \((\(-1\) + x)\)\^4\ \((1 + x)\)\^5\ \((\(-1\) - 2\ x + x\^2)\)\ \((\(-1\) + 2\ x + 
                      x\^2)\)\ \((1 - 5\ x - x\^2 + x\^3)\)\^2\ \((\(-1\) + 6\ x + 21\ x\^2 - 12\ x\^3 - 9\ x\^4 + 2\ x\^5 + x\^6)\)\^2, {"gP[16,-8]"}}, 
{"3,32,8,1", \((\(-1\) + x)\)\^2\ \((1 + x)\)\ \((\(-3\) + x\^2)\)\^2\ \((3 + 4\ x - 4\ x\^2 - 2\ x\^3 + 
                      x\^4)\)\ \((\(-1\) + 8\ x - 4\ x\^2 - 2\ x\^3 + 
                      x\^4)\)\ \((1 - 20\ x + 14\ x\^2 + 48\ x\^3 - 6\ x\^4 - 28\ x\^5 - 4\ x\^6 + 4\ x\^7 + x\^8)\)\^2, {{{\(-4\), 16}, 
                    
                    16}}}, 
{"3,32,8,2", \((\(-1\) + x)\)\^9\ \((1 + x)\)\^9\ \((3 + 
                      x)\)\ \((\(-5\) + x\^2)\)\^2\ \((17 - 10\ x\^2 + x\^4)\)\^2, {{{\(-15\), 15}, 
                    16}}}, 
{"3,32,,2", \((\(-1\) + x)\)\^3\ \((1 + x)\)\^3\ \((3 + 
                      x)\)\ \((\(-1\) - 2\ x + x\^2)\)\^2\ \((\(-1\) + 2\ x + x\^2)\)\^2\ \((7 - 6\ x\^2 + x\^4)\)\^4, {{{\(-13\), 13}, 
                    16}, "gP[16,7]"}}, 
{"3,32", \((\(-1\) + x)\)\^3\ \((1 + x)\)\^3\ \((3 + 
                      x)\)\ \((\(-3\) + x\^2)\)\^4\ \((17 - 52\ x\^2 + 42\ x\^4 - 12\ x\^6 + x\^8)\)\^2, {{{\(-11\), 11}, 
                    16}}}, 
{"3,32,4", \((\(-1\) + x)\)\^5\ \((1 + x)\)\^5\ \((3 + 
                      x)\)\ \((\(-5\) + x\^2)\)\^2\ \((17 - 60\ x\^2 + 46\ x\^4 - 12\ x\^6 + x\^8)\)\^2, {{{\(-7\), 7}, 
                    16}}}, 
{"3,32,4", \((\(-1\) + x)\)\^2\ \((1 + 
                      x)\)\ \((\(-1\) - 2\ x + x\^2)\)\^2\ \((3 - 6\ x\^2 + x\^4)\)\^2\ \((\(-1\) - 16\ x + 50\ x\^2 + 80\ x\^3 - 6\ x\^4 - 36\ x\^5 - 
                      6\ x\^6 + 4\ x\^7 + x\^8)\)\ \((\(-1\) - 8\ x + 
                      6\ x\^2 + 24\ x\^3 - 6\ x\^4 - 20\ x\^5 - 2\ x\^6 + 
                      4\ x\^7 + x\^8)\), {{{8, 16, 16, \(-8\)}, 8}}}\)},{},
{  {"3,34,1", -20 - 600 x - 4388 (x^2) + 
                  6464 (x^3) + 118952 (x^4) + 19092 (x^5) - 1133273 (x^6) - 
                  379841 (x^7) + 5220029 (x^8) + 1790735 (x^9) - 
                  13578320 (x^10) - 4559772 (x^11) + 21994105 (x^12) + 
                  7289853 (x^13) - 23663400 (x^14) - 7810998 (x^15) + 
                  17635767 (x^16) + 5824979 (x^17) - 9342429 (x^18) - 
                  3093043 (x^19) + 3564147 (x^20) + 1182993 (x^21) - 
                  980860 (x^22) - 326222 (x^23) + 192674 (x^24) + 
                  64166 (x^25) - 26304 (x^26) - 8766 (x^27) + 2367 (x^28) + 
                  789 (x^29) - 126 (x^30) - 42 (x^31) + 3 (x^32) + 
                  (x^33), {{{7, -5, 16, 5, -7}, 13}}}},{},
{  \({"3,36,72", \((\(-2\) + x)\)\^2\ x\^2\ \((1 + x)\)\^3\ \((\(-2\) + 2\ x + x\^2)\)\^2\ \((1 - 3\ x\^2 + x\^3)\)\^2\ \((3 - 3\ x\^2 + x\^3)\)\^2\ \((1 - 12\ x\^2 - 4\ x\^3 + 9\ x\^4 + 6\ x\^5 + x\^6)\)\^2, {{{18}, 
                    18}}}, 
{"3,36,24", \((\(-2\) + 
                      x)\)\ x\ \((1 + x)\)\^4\ \((\(-3\) + x\^2)\)\^2\ \((\(-2\) + 2\ x + x\^2)\)\^2\ \((4 - 6\ x - x\^2 + 
                      x\^3)\)\ \((1 - 3\ x - x\^2 + x\^3)\)\^2\ \((\(-11\) - 6\ x + 21\ x\^2 + 2\ x\^3 - 9\ x\^4 + x\^6)\)\^2, {"gP[-6,6]"}}, {"3,36,18", \((\(-2\) + x)\)\^2\ \((1 + x)\)\^3\ \((1 - 3\ x + x\^3)\)\^2\ \((3 - 5\ x - x\^2 + x\^3)\)\^2\ \((\(-3\) + 3\ x\^2 + x\^3)\)\^2\ \((1 + 9\ x + 18\ x\^2 - x\^3 - 9\ x\^4 + x\^6)\)\^2, {"gS[3,{{.2,9,3,.43},{1.,9,0},{.48,9,2,.5},{.75,-9,1,-.44}}]"}}\)},{},
{  \({"F38", \((3 + 
                      x)\)\ \((\(-7\) + 15\ x\^2 - 8\ x\^4 + x\^6)\)\^6, {{{15, \(-15\)}, 19}}}, 
{"3,38,,1", 
                172 - 2140\ x - 21584\ x\^2 + 98020\ x\^3 + 685292\ x\^4 - 
                  1224012\ x\^5 - 7549897\ x\^6 + 5978151\ x\^7 + 
                  40682877\ x\^8 - 11906237\ x\^9 - 124463914\ x\^10 + 
                  3855218\ x\^11 + 240609969\ x\^12 + 27851253\ x\^13 - 
                  315338619\ x\^14 - 63811319\ x\^15 + 293295506\ x\^16 + 
                  74808476\ x\^17 - 199408768\ x\^18 - 57337758\ x\^19 + 
                  100927931\ x\^20 + 31034411\ x\^21 - 38387924\ x\^22 - 
                  12266028\ x\^23 + 10989687\ x\^24 + 3588563\ x\^25 - 
                  2352624\ x\^26 - 777282\ x\^27 + 370461\ x\^28 + 
                  123107\ x\^29 - 41602\ x\^30 - 13858\ x\^31 + 3150\ x\^32 + 
                  1050\ x\^33 - 144\ x\^34 - 48\ x\^35 + 3\ x\^36 + 
                  x\^37, {{{\(-9\), \(-5\), 16, 11, \(-11\), \(-16\), 10, 
                      16}, 8}}}\)},{},
{  \({"F40 480", \((\(-2\) + x)\)\^4\ \((\(-1\) + x)\)\^5\ x\^8\ \((1 + x)\)\^5\ \((2 + x)\)\^4\ \((3 + 
                      x)\)\ \((\(-5\) + x\^2)\)\^6, {{{15, 
                      9, \(-9\), \(-15\)}, 
                    10}}}, 
{"3,40,80", \((\(-1\) + x)\)\^3\ \((1 + x)\)\^3\ \((3 + 
                      x)\)\ \((1 - 3\ x + x\^2)\)\^2\ \((\(-1\) - x + x\^2)\)\^2\ \((\(-1\) + x + x\^2)\)\^2\ \((1 + 3\ x + x\^2)\)\^2\ \((11 - 7\ x\^2 + x\^4)\)\^4, {{{\(-13\), \(-11\), 11, 13}, 
                    7}}}, {"3,40,20", \((\(-1\) + x)\)\ \((\(-5\) + 
                      x\^2)\)\ \((17 - 10\ x\^2 + 
                      x\^4)\)\ \((\(-1\) + 10\ x - 25\ x\^2 - 5\ x\^3 + 29\ x\^4 - 10\ x\^6 + x\^8)\)\^2\ \((5 - 10\ x - 15\ x\^2 + 25\ x\^3 + 19\ x\^4 - 14\ x\^5 - 8\ x\^6 + 2\ x\^7 + x\^8)\)\^2, {"gS[3,{{5.5,5,1,.945},{4,10,1,.95},{3,10,1,-2.19},{2,10,1,.95},{1,5,1,-2.2}}]"}}, {"Flower-
NonSnark 120", \((\(-2\) + x)\)\^2\ \((\(-1\) + 
                      x)\)\ \((1 + x)\)\ \((2 + x)\)\^2\ \((3 + 
                      x)\)\ \((\(-1\) - x + x\^2)\)\^4\ \((\(-1\) + x + x\^2)\)\^4\ \((9 + 3\ x - 7\ x\^2 - x\^3 + x\^4)\)\^2\ \((9 - 3\ x - 7\ x\^2 + x\^3 + x\^4)\)\^2, {"gS[3,{{3,20,0,.156},{2,10,0},{1,10,1}},{{1,3,-1},{2,1,1}}]"}}\)},{},
{  \({"F42", \((3 + 
                      x)\)\ \((\(-3\) + x\^2)\)\^2\ \((\(-1\) + 84\ x\^2 - 1028\ x\^4 + 3870\ x\^6 - 5164\ x\^8 + 3356\ x\^10 - 1189\ x\^12 + 234\ x\^14 - 24\ x\^16 + x\^18)\)\^2, {{{17, 13, \(-13\), 7, \(-7\), \(-17\)}, 
                    7}}}, {"3,42,28", \((\(-1\) + x)\)\ \((\(-7\) + 
                      x\^2)\)\ \((\(-3\) + 
                      x\^2)\)\ \((1 - 15\ x + 53\ x\^2 - 19\ x\^3 - 56\ x\^4 + 29\ x\^5 + 19\ x\^6 - 10\ x\^7 - 2\ x\^8 + x\^9)\)\^2\ \((7 - 7\ x - 35\ x\^2 + 21\ x\^3 + 56\ x\^4 - 7\ x\^5 - 29\ x\^6 - 4\ x\^7 + 4\ x\^8 + x\^9)\)\^2, {"gS[3,{{1,7},{.8,14},{.6,14,1,3.1},{.4,7}}]"}}, 
{"3,42,1", 
                63 + 737\ x - 10570\ x\^2 - 49672\ x\^3 + 478745\ x\^4 + 
                  1113361\ x\^5 - 8783703\ x\^6 - 11178771\ x\^7 + 
                  76109484\ x\^8 + 62997726\ x\^9 - 355983235\ x\^10 - 
                  222985899\ x\^11 + 1009086143\ x\^12 + 527961425\ x\^13 - 
                  1882146838\ x\^14 - 872624288\ x\^15 + 2444625491\ x\^16 + 
                  1040758797\ x\^17 - 2299119385\ x\^18 - 919276195\ x\^19 + 
                  1608092494\ x\^20 + 613124642\ x\^21 - 851391594\ x\^22 - 
                  312951312\ x\^23 + 344784550\ x\^24 + 123200986\ x\^25 - 
                  107236046\ x\^26 - 37494698\ x\^27 + 25559452\ x\^28 + 
                  8790972\ x\^29 - 4625159\ x\^30 - 1571595\ x\^31 + 
                  623692\ x\^32 + 210110\ x\^33 - 60658\ x\^34 - 
                  20318\ x\^35 + 4017\ x\^36 + 1341\ x\^37 - 162\ x\^38 - 
                  54\ x\^39 + 3\ x\^40 + 
                  x\^41, {{{19, 13, \(-7\), 8, 7, \(-13\), \(-19\)}, 6}}}\)},{},
{  \({"3,44,", \((\(-1\) + x)\)\ \((\(-5\) + 
                      x\^2)\)\ \((1 - 7\ x + 13\ x\^2 - 5\ x\^3 - 2\ x\^4 + x\^5)\)\^2\ \((\(-1\) - 7\ x - 13\ x\^2 - 5\ x\^3 + 2\ x\^4 + x\^5)\)\^2\ \((23 + 74\ x + 17\ x\^2 - 132\ x\^3 - 73\ x\^4 + 84\ x\^5 + 49\ x\^6 - 22\ x\^7 - 12\ x\^8 + 2\ x\^9 + x\^10)\)\^2, {"gP[-11,-11]"}}, {"3,44,22", \((\(-2\) + x)\)\^2\ \((1 + 
                      x)\)\ \((\(-1\) + 53\ x - 522\ x\^2 + 1364\ x\^3 + 1736\ x\^4 - 8706\ x\^5 - 2508\ x\^6 + 19818\ x\^7 + 4170\ x\^8 - 21311\ x\^9 - 5072\ x\^10 + 12287\ x\^11 + 3320\ x\^12 - 4025\ x\^13 - 1186\ x\^14 + 750\ x\^15 + 234\ x\^16 - 74\ x\^17 - 24\ x\^18 + 3\ x\^19 + x\^20)\)\^2, {"gS[3,{{.3,11,3,.53},{1.,11,0},{.58,11,2,.5},{.845,-11,1,-.33}}]"}}\)},{},
{  \({"3,46,", \((\(-1\) + 
                      
                      x)\)\ \((1 + 16\ x - 34\ x\^2 - 1360\ x\^3 - 2939\ x\^4 + 16231\ x\^5 + 16559\ x\^6 - 52149\ x\^7 - 38643\ x\^8 + 69416\ x\^9 + 44960\ x\^10 - 48415\ x\^11 - 29204\ x\^12 + 19678\ x\^13 + 11321\ x\^14 - 4846\ x\^15 - 2684\ x\^16 + 714\ x\^17 + 382\ x\^18 - 58\ x\^19 - 30\ x\^20 + 2\ x\^21 + x\^22)\)\^2, {"gP[23,9]"}}\)},{},
{  \({"F48 288", \((\(-2\) + x)\)\^6\ \((\(-1\) + x)\)\^3\ x\^12\ \((1 + x)\)\^3\ \((2 + x)\)\^6\ \((3 + 
                      x)\)\ \((\(-6\) + x\^2)\)\^4\ \((\(-3\) + x\^2)\)\^4, {{{\(-7\), 9, 19, \(-19\), \(-9\), 7}, 
                    8}}}, 
{"GreatRhombi-
cuboctahedral 48", \((\(-2\) + x)\)\^2\ \((\(-1\) + x)\)\^4\ x\^4\ \((1 + x)\)\^4\ \((2 + x)\)\^2\ \((3 + 
                      x)\)\ \((\(-2\) - 2\ x + x\^2)\)\^3\ \((\(-2\) + 2\ x + x\^2)\)\^3\ \((2 - 4\ x - x\^2 + x\^3)\)\^3\ \((\(-2\) - 4\ x + x\^2 + x\^3)\)\^3, {"gS[3,{{4,8,1,-.393},{3,16,1,-.2,-1},{2,16,1,-.17,-1},{1,8,1,-.34,-1}},{{1,9,2,4,4},{2,12,2,4,4},{10,26,4,4,4},{11,27,4,4,4},{25,41,4,2,4},
{28,42,4,2,4}}]"}}, 
{"3,48,32", \((\(-2\) + x)\)\ x\ \((\(-1\) - 2\ x + x\^2)\)\^2\ \((4 - 6\ x - x\^2 + 
                      x\^3)\)\ \((\(-2\) - 4\ x + x\^2 + x\^3)\)\^2\ \((2 - 4\ x\^2 + x\^4)\)\^2\ \((\(-34\) - 40\ x + 228\ x\^2 + 352\ x\^3 - 222\ x\^4 - 472\ x\^5 + 44\ x\^6 + 240\ x\^7 + 23\ x\^8 - 52\ x\^9 - 10\ x\^10 + 4\ x\^11 + x\^12)\)\^2, {"gP[-8,8]"}}, 
{"3,48,8,1", \((\(-1\) + x)\)\^2\ \((1 + x)\)\ \((\(-7\) + x\^2)\)\ \((\(-3\) + 
                      x\^2)\)\ \((\(-1\) + 2\ x + x\^2)\)\^2\ \((1 - 3\ x - x\^2 + x\^3)\)\^2\ \((\(-1\) - 3\ x + x\^2 + x\^3)\)\^2\ \((1 - 18\ x + 11\ x\^2 + 12\ x\^3 - 7\ x\^4 - 2\ x\^5 + x\^6)\)\^2\ \((\(-7\) + 10\ x + 11\ x\^2 - 12\ x\^3 - 7\ x\^4 + 2\ x\^5 + x\^6)\)\^2, {"gS[3,{{1,8},{.8,16},{.6,16,1,3.1},{.4,8}}]"}}, {"3,48,8,2", \((\(-2\) + x)\)\^4\ \((\(-1\) + x)\)\^2\ x\^10\ \((1 + x)\)\^2\ \((2 + x)\)\^4\ \((3 + x)\)\ \((\(-4\) - x + x\^2)\)\ \((\(-4\) + x + 
                      x\^2)\)\ \((\(-72\) + 356\ x\^2 - 328\ x\^4 + 117\ x\^6 - 18\ x\^8 + x\^10)\)\^2, {{{\(-7\), \(-3\), 19, \(-19\), \(-9\), 7, \(-7\), 
                      9, 19, \(-19\), 3, 7}, 4}}}\)},{},
{  \({"3,50,,1", \((3 + 
                      x)\)\ \((\(-1\) - x + x\^2)\)\^6\ \((\(-1\) + x + x\^2)\)\^6\ \((11 - 8\ x\^2 + x\^4)\)\^6, {{{21, \(-21\), \(-19\), 19, \(-19\), 
                      19, \(-19\), 19, 21, \(-21\)}, 
                    5}}}, 
{"3,50,,2", \((\(-1\) + 
                      x)\)\ \((\(-1\) - x + x\^2)\)\^2\ \((1 + 3\ x + x\^2)\)\^2\ \((49 - 280\ x - 2165\ x\^2 + 1805\ x\^3 + 14165\ x\^4 - 4204\ x\^5 - 35520\ x\^6 + 3870\ x\^7 + 42445\ x\^8 - 1720\ x\^9 - 28028\ x\^10 + 395\ x\^11 + 11020\ x\^12 - 45\ x\^13 - 2645\ x\^14 + 2\ x\^15 + 380\ x\^16 - 30\ x\^18 + x\^20)\)\^2, {"gP[25,9]"}}, 
{"3,50,,3", \((1 + x)\)\ \((\(-1\) + 2\ x + x\^2)\)\ \((5 - x - 3\ x\^2 + x\^3)\)\ \((1 - 
                      3\ x - x\^2 + 
                      x\^3)\)\ \((\(-5\) - 65\ x + 54\ x\^2 + 1831\ x\^3 - 330\ x\^4 - 10115\ x\^5 - 1433\ x\^6 + 21318\ x\^7 + 6372\ x\^8 - 21522\ x\^9 - 8209\ x\^10 + 11680\ x\^11 + 5014\ x\^12 - 3570\ x\^13 - 1638\ x\^14 + 613\ x\^15 + 293\ x\^16 - 55\ x\^17 - 27\ x\^18 + 2\ x\^19 + x\^20)\)\^2, {"gS[3,{{7,15,1,-.315},{10,5,1,-.315},{5,20,1,-.175,-1},{3,5,0,-.315},{1,5,1,-.315}},{{2,23,3,4,5},{3,24,3,4,5}}]"}}, {"3,50,20", \((\(-1\) + 
                      x)\)\ \((31 - 12\ x\^2 + x\^4)\)\ \((11 - 8\ x\^2 + 
                      x\^4)\)\ \((4 - 26\ x + 27\ x\^2 + 56\ x\^3 - 67\ x\^4 - 41\ x\^5 + 47\ x\^6 + 11\ x\^7 - 12\ x\^8 - x\^9 + x\^10)\)\^2\ \((\(-4\) + 18\ x + 7\ x\^2 - 72\ x\^3 - 13\ x\^4 + 73\ x\^5 + 17\ x\^6 - 27\ x\^7 - 8\ x\^8 + 3\ x\^9 + x\^10)\)\^2, {"gS[3,{{6.5,5,1,.945},{5,10,1,.95},{4,10,1,-2.19},{3,10,1,.95},{2,10,1,-2.2},{1,5,1,.95}}]"}}\)},{},
{  \({"3,52,,1", \((\(-1\) + 
                      x)\)\ x\^14\ \((2 + x)\)\^14\ \((2131 + 3821\ x - 1047\ x\^2 - 4504\ x\^3 - 143\ x\^4 + 2295\ x\^5 + 132\ x\^6 - 636\ x\^7 + 95\ x\^9 - 8\ x\^10 - 6\ x\^11 + x\^12)\)\^2\ \((13 + 91\ x + 91\ x\^2 - 208\ x\^3 - 221\ x\^4 + 221\ x\^5 + 170\ x\^6 - 136\ x\^7 - 50\ x\^8 + 45\ x\^9 + 2\ x\^10 - 6\ x\^11 + x\^12)\)\^2, {"gP[26,13]"}}\)},{},
{  \({"F54", 
                x\^4\ \((3 + 
                      x)\)\ \((\(-3\) + x\^2)\)\^6\ \((3 - 3\ x\^2 + x\^3)\)\^6\ \((\(-3\) + 3\ x\^2 + x\^3)\)\^6, {{{\(-13\), \(-11\), 
                      11, \(-11\), 11, 13}, 
                    9}}}, 
{"3,54,,1", \((\(-1\) + 
                      x)\)\ x\^2\ \((2 + x)\)\^2\ \((\(-3\) - 9\ x + 21\ x\^2 + 2\ x\^3 - 9\ x\^4 + x\^6)\)\^2\ \((\(-3\) + 189\ x - 1458\ x\^2 + 3789\ x\^3 - 1431\ x\^4 - 8307\ x\^5 + 8895\ x\^6 + 6138\ x\^7 - 10233\ x\^8 - 2050\ x\^9 + 5625\ x\^10 + 315\ x\^11 - 1719\ x\^12 - 18\ x\^13 + 297\ x\^14 - 27\ x\^16 + x\^18)\)\^2, {"gP[27,13]"}}, 
{"3,54,,2", \((\(-1\) + 
                      x)\)\ \((\(-3\) - x + x\^2)\)\^2\ \((3 + 18\ x - 48\ x\^2 + 23\ x\^3 + 6\ x\^4 - 6\ x\^5 + x\^6)\)\^2\ \((3 + 27\ x - 540\ x\^3 - 1269\ x\^4 + 819\ x\^5 + 5262\ x\^6 + 4131\ x\^7 - 3213\ x\^8 - 5872\ x\^9 - 1206\ x\^10 + 2277\ x\^11 + 1374\ x\^12 - 135\ x\^13 - 324\ x\^14 - 69\ x\^15 + 18\ x\^16 + 9\ x\^17 + x\^18)\)\^2, {"gP[27,9]"}}, {"3,54,36", \((\(-2\) + x)\)\ \((\(-1\) + x)\)\^2\ x\^2\ \((1 + x)\)\^2\ \((2 + 
                      x)\)\ \((3 + 
                      x)\)\ \((\(-1\) - 3\ x + x\^3)\)\^2\ \((1 - 3\ x + x\^3)\)\^2\ \((3 - 6\ x\^2 + x\^4)\)\^2\ \((3 - 72\ x\^2 + 342\ x\^4 - 411\ x\^6 + 150\ x\^8 - 21\ x\^10 + x\^12)\)\^2, {"gS[3,{{3,18,1},{2,18,0},{1,18,1}},{{19,1,1},{2,18,-17}}]"}}, {"3,54,", \((3 + 
                      x)\)\ \((\(-3\) + x\^2)\)\^2\ \((\(-3\) + 9\ x\^2 - 6\ x\^4 + x\^6)\)\^2\ \((\(-111\) + 1539\ x\^2 - 6705\ x\^4 + 12087\ x\^6 - 11115\ x\^8 + 5742\ x\^10 - 1725\ x\^12 + 297\ x\^14 - 27\ x\^16 + x\^18)\)\^2, {"gS[3,{{4,18,0,-.874,-1},{2,18,0,0,-1},{1,18}},{{1,19,2,2,9},{2,20,2,2,9},{1,23,2,2,8},{2,24,2,2,7},{15,19,2,2,2},{16,20,2,2,2},{1,2,2,2,9}}]"}}, 
{"Gray", 
                x\^16\ \((3 + 
                      x)\)\ \((\(-6\) + x\^2)\)\^6\ \((\(-3\) + x\^2)\)\^12, {{{13, \(-13\), 7, \(-7\), 25, \(-25\)}, 
                    9}, "gS[3,{{1,9,0,-1.94},{2.,9,0,.13,-1},{3,27,0,.38,-1},{3.5,9,0,.15,18}},{{46,21,1,3,9},{10,19,0,3,3},{11,20,0,3,3},{12,21,0,3,3},{13,28,0,3,3},{14,29,0,3,3},{15,30,0,3,3},{16,37,0,3,3},{17,38,0,3,3},{18,39,0,3,3},{1,19,0,9,3},{2,20,0,9,3},{3,21,0,9,3},{4,22,0,9,3},{5,23,0,9,3},{6,24,0,9,3},{7,25,0,9,3},{8,26,0,9,3},{9,27,0,9,3}}]" (*Edge - 
                      transitive\ but\ not\ vertex - 
                      transitive\ [6, p119] . \ 
                          3\[Cross]3\[Cross]3\ block\ of\ 27\ xyz\ cubes\ plus\ 27\ xy0\ "lines", \ 
                    cubes\ joined\ to\ parent\ lines . \ Vertex\ indices\ x + 
                      3  y - 3, \ x + 3  z + 6, \ 
                    y + 3  z + 42\ for\ lines, \ 
                    x + 3  y + 9  z + 6\ for\ \(\(cubes\)\(.\)\)\ *) }}\)},{},
{  \({"F56a", \((\(-1\) + x)\)\^3\ \((1 + x)\)\^3\ \((3 + x)\)\ \((\(-2\) + x\^2)\)\^6\ \((\(-8\) + 24\ x\^2 - 10\ x\^4 + x\^6)\)\^6, {{{\(-13\), \(-11\), 11, 13}, 
                    14}, "gS[3,{{2,28,0},{3,28}},{{9,19,11,17,11,17,9,19,0,0},20}]"}}, 
{"3,56,,1", \((\(-1\) + x)\)\ \((1 + 
                      x)\)\ \((3 + x)\)\ \((\(-1\) - 2\ x + 
                      x\^2)\)\ \((\(-1\) + 2\ x + 
                      
                      x\^2)\)\ \((8 - 264\ x\^2 + 3114\ x\^4 - 17231\ x\^6 + 49960\ x\^8 - 81946\ x\^10 + 79400\ x\^12 - 46537\ x\^14 + 16741\ x\^16 - 3686\ x\^18 + 481\ x\^20 - 34\ x\^22 + x\^24)\)\^2, {{{\(-9\), \(-11\), \(-13\), \(-15\), 15, 13, 11, 9}, 
                    5}}}, 
{"3,56,,2", \((1 + 
                      x)\)\ \((1 + 3\ x - 4\ x\^2 + x\^3)\)\ \((1 - x - 
                      2\ x\^2 + x\^3)\)\ \((\(-1\) - x + 2\ x\^2 + 
                      x\^3)\)\ \((\(-1\) + 3\ x + 4\ x\^2 + 
                      x\^3)\)\ \((\(-1\) - 9\ x + 3\ x\^2 + 23\ x\^3 - 6\ x\^4 - 10\ x\^5 + x\^6 + x\^7)\)\^2\ \((\(-5\) + 89\ x\^2 - 389\ x\^4 + 
                      671\ x\^6 - 498\ x\^8 + 156\ x\^10 - 21\ x\^12 + 
                      x\^14)\)\ \((\(-1\) + 61\ x\^2 - 389\ x\^4 + 
                      699\ x\^6 - 498\ x\^8 + 156\ x\^10 - 21\ x\^12 + 
                      x\^14)\), {{{\(-8\), \(-10\), \(-12\), \(-14\), \(-16\), \(-18\), \(-20\), 20, 18, 16, 14, 12, 10, 8}, 
                    3}}}, {"3,56,,3", \((\(-1\) + x)\)\^3\ \((1 + x)\)\^3\ \((3 + x)\)\ \((\(-8\) + 24\ x\^2 - 10\ x\^4 + x\^6)\)\^2\ \((\(-463\) + 3532\ x\^2 - 9956\ x\^4 + 14160\ x\^6 - 11460\ x\^8 + 5550\ x\^10 - 1625\ x\^12 + 280\ x\^14 - 26\ x\^16 + x\^18)\)\^2, {"gS[3,{{.85,28,0},{1,28}},{{1,11,-3},{2,-9,5},{2,-7,-1},{3,13,-5}}]"}}\)},{},
{  \({"3,58", \((\(-1\) + 
                      x)\)\ \((\(-59\) - 629\ x + 1174\ x\^2 + 22118\ x\^3 + 6279\ x\^4 - 208797\ x\^5 - 123758\ x\^6 + 831046\ x\^7 + 542553\ x\^8 - 1740414\ x\^9 - 1133697\ x\^10 + 2164116\ x\^11 + 1368832\ x\^12 - 1711763\ x\^13 - 1045448\ x\^14 + 896822\ x\^15 + 529492\ x\^16 - 317850\ x\^17 - 181888\ x\^18 + 76525\ x\^19 + 42541\ x\^20 - 12328\ x\^21 - 6668\ x\^22 + 1272\ x\^23 + 670\ x\^24 - 76\ x\^25 - 39\ x\^26 + 2\ x\^27 + x\^28)\)\^2, {"gP[29,11]"}}\)},{},
{  \({"F60", \((\(-1\) + x)\)\^13\ x\^4\ \((2 + x)\)\^4\ \((\(-5\) + x\^2)\)\^3\ \((\(-3\) - x + x\^2)\)\^10\ \((1 + 3\ x + x\^2)\)\^6, {{{12, \(-17\), \(-12\), 25, 17, \(-26\), \(-9\), 9, \(-25\), 
                      26}, 6}}}, {"3,60,,1", \((\(-1\) + 
                      x)\)\ \((\(-7\) + x\^2)\)\ \((\(-5\) + 
                      x\^2)\)\ \((\(-3\) + 
                      x\^2)\)\ \((\(-1\) - x + x\^2)\)\^2\ \((13 - 10\ x\^2 + 
                      x\^4)\)\ \((\(-9\) + 33\ x + 11\ x\^2 - 100\ x\^3 - 18\ x\^4 + 91\ x\^5 + 22\ x\^6 - 30\ x\^7 - 9\ x\^8 + 3\ x\^9 + x\^10)\)\^2\ \((\(-5\) + 40\ x - 75\ x\^2 - 70\ x\^3 + 209\ x\^4 + 35\ x\^5 - 195\ x\^6 - 5\ x\^7 + 81\ x\^8 - 15\ x\^10 + x\^12)\)\^2, {"gS[3,{{1.8,5,1},{1.3,10,1},{1.1,10,1,3.1},{.9,10},{.7,10,1,3.1},{.5,10,1},{.3,5,1,3.1}}]"}}, 
{"Truncated
Icosahedral", \((\(-1\) + x)\)\^9\ \((2 + x)\)\^4\ \((\(-3\) - x + x\^2)\)\^5\ \((\(-4\) + x + x\^2)\)\^4\ \((\(-1\) + x + x\^2)\)\^5\ \((1 + 3\ x + x\^2)\)\^3\ \((1 + 7\ x - 2\ x\^2 - 3\ x\^3 + x\^4)\)\^3, {"gS[3,{{1.8,5,1,-.314},{1.3,15,1,0,-1},{1,20,1,.15,-1},{.65,15,1,-.15,-1},{.3,5,1,-2.2}},{{1,7,1,3,5},{21,41,4,3,5},{24,42,4,3,5},{6,22,3,4,5},{8,23,3,4,5}}]"}}, {"3,60,20", \((\(-1\) + x)\)\ \((\(-3\) - x + 
                      x\^2)\)\ \((\(-1\) + x + x\^2)\)\ \((1 + 3\ x + 
                      x\^2)\)\ \((1 + 7\ x - 2\ x\^2 - 3\ x\^3 + 
                      x\^4)\)\ \((1 - 3\ x - 89\ x\^2 + 84\ x\^3 + 336\ x\^4 - 134\ x\^5 - 309\ x\^6 + 72\ x\^7 + 114\ x\^8 - 15\ x\^9 - 18\ x\^10 + x\^11 + x\^12)\)\^2\ \((11 - 11\ x - 87\ x\^2 + 60\ x\^3 + 218\ x\^4 - 90\ x\^5 - 215\ x\^6 + 56\ x\^7 + 90\ x\^8 - 13\ x\^9 - 16\ x\^10 + x\^11 + x\^12)\)\^2, {"gS[3,{{6.5,5,1,.945},{5,15,1,1.37},{4,20,1,-2.7},{3,15,1,-2.6},{2,5,1,1.6}}]"}}\)},{},
{  \({"F62", \((3 + 
                      x)\)\ \((\(-25\) + 100\ x\^2 - 129\ x\^4 + 66\ x\^6 - 14\ x\^8 + x\^10)\)\^6, {{{11, \(-11\)}, 
                    31}}}, 
{"3,62,?,1", \((\(-1\) + 
                      x)\)\ \((\(-32\) - 1840\ x - 13904\ x\^2 - 13800\ x\^3 + 152110\ x\^4 + 366229\ x\^5 - 493178\ x\^6 - 1986870\ x\^7 + 369049\ x\^8 + 5060951\ x\^9 + 1111966\ x\^10 - 7356769\ x\^11 - 2949988\ x\^12 + 6758699\ x\^13 + 3309158\ x\^14 - 4162219\ x\^15 - 2215270\ x\^16 + 1774900\ x\^17 + 975360\ x\^18 - 531257\ x\^19 - 293408\ x\^20 + 111289\ x\^21 + 60849\ x\^22 - 15969\ x\^23 - 8572\ x\^24 + 1494\ x\^25 + 784\ x\^26 - 82\ x\^27 - 42\ x\^28 + 2\ x\^29 + x\^30)\)\^2, {"gP[31,11]"}}\)},{},
{  \({"F64", \((\(-1\) + x)\)\^9\ \((1 + x)\)\^9\ \((3 + x)\)\ \((\(-5\) + x\^2)\)\^6\ \((\(-3\) + x\^2)\)\^4\ \((\(-1\) - 2\ x + x\^2)\)\^6\ \((\(-1\) + 2\ x + x\^2)\)\^6, {{{23, \(-11\), \(-29\), 
                      25, \(-25\), 29, 11, \(-23\)}, 
                    8}}}, 
{"3,64, ,1", \((\(-1\) + 
                      x)\)\ \((1 + x)\)\ \((3 + x)\)\ \((\(-1\) - 2\ x + 
                      x\^2)\)\ \((\(-1\) + 2\ x + 
                      x\^2)\)\ \((1 - 4\ x\^2 + x\^4)\)\^2\ \((1 - 28\ x\^2 + 40\ x\^4 - 12\ x\^6 + x\^8)\)\^2\ \((289 - 2176\ x\^2 + 5260\ x\^4 - 5952\ x\^6 + 3592\ x\^8 - 1224\ x\^10 + 236\ x\^12 - 24\ x\^14 + x\^16)\)\^2, {"gS[3,{{1,64}},{{1,23,-11},{3,-29,25}}]"}}, 
{"3,64,?,2", \((\(-1\) + x)\)\^8\ \((1 + x)\)\^9\ \((\(-1\) - 2\ x + x\^2)\)\ \((\(-1\) + 2\ x + 
                      x\^2)\)\ \((1 - 5\ x - x\^2 + x\^3)\)\^2\ \((\(-1\) - 6\ x + 21\ x\^2 + 12\ x\^3 - 9\ x\^4 - 2\ x\^5 + x\^6)\)\^2\ \((\(-1\) - 20\ x - 18\ x\^2 + 268\ x\^3 + 313\ x\^4 - 616\ x\^5 - 212\ x\^6 + 312\ x\^7 + 75\ x\^8 - 60\ x\^9 - 14\ x\^10 + 4\ x\^11 + x\^12)\)\^2, {"gP[32,-16]"}}\)},{},
{  {"", 0, {}}},{},
{  {"", 0, {}}},{},
{  \({"Harries
Cage3,10 120", , {{{\(-35\), \(-27\), 
                      27, \(-23\), 15, \(-15\), \(-9\), \(-35\), 23, \(-27\), 
                      27, 9, 15, \(-15\)}, 
                    5}, {{\(-35\), 9, 15, \(-15\), 23, \(-27\), 27, \(-35\), 
                      15, \(-15\), \(-9\), \(-27\), 27, \(-23\)}, 
                    5}, {{\(-33\), \(-27\), 27, \(-21\), \(-13\), 13, 19, 
                      29, \(-29\), 33, 21, \(-19\), \(-13\), 13}, 
                    5}, {{\(-33\), \(-27\), 27, 9, \(-13\), 13, \(-21\), 
                      29, \(-29\), 33, \(-13\), 13, \(-9\), 21}, 
                    5}}}, {"Balaban
Cage10,3 80", \((\(-2\) + 
                      x)\)\ \((\(-1\) + x)\)\^8\ x\^2\ \((1 + x)\)\^8\ \((2 + 
                      x)\)\ \((3 + 
                      x)\)\ \((\(-6\) + x\^2)\)\^2\ \((\(-5\) + x\^2)\)\^4\ \((\(-2\) + x\^2)\)\^2\ \((3 - 6\ x\^2 + x\^4)\)\^8, {"gS[3,{{1,50},{.5,20,6,-.11,1}},{{3,28,5,5,5},{1,10,5,5,10},{5,41,1}}]"}}\)},{},
{  \({"F72", \((\(-2\) + x)\)\^6\ \((\(-1\) + x)\)\^15\ x\^4\ \((1 + x)\)\^15\ \((2 + x)\)\^6\ \((3 + 
                      x)\)\ \((\(-7\) + x\^2)\)\^6\ \((\(-3\) + x\^2)\)\^6, {{{5, \(-9\), 31, \(-31\), 9, \(-5\)}, 12}}}\)},{},
{  \({"F74", \((3 + 
                      x)\)\ \((47 - 241\ x\^2 + 402\ x\^4 - 298\ x\^6 + 105\ x\^8 - 17\ x\^10 + x\^12)\)\^6, {{{21, \(-21\)}, 37}}}\)},{},
{  {"76", 0, {}}},{},
{  \({"78", 
                x\^4\ \((3 + 
                      x)\)\ \((3 - 5\ x\^2 + x\^4)\)\^6\ \((39 - 78\ x\^2 + 52\ x\^4 - 13\ x\^6 + x\^8)\)\^6, {{{33, \(-33\)}, 39}}}\)},{},
{  \({"80", \((\(-2\) + x)\)\^4\ \((\(-1\) + x)\)\^17\ x\^8\ \((1 + x)\)\^17\ \((2 + x)\)\^4\ \((3 + 
                      x)\)\ \((\(-6\) + x\^2)\)\^8\ \((\(-5\) + x\^2)\)\^6, {{{\(-9\), 25, \(-25\), 9}, 
                    20}}}, {"Goldberg", \((\(-2\) + x)\)\ x\^2\ \((2 + 
                      x)\)\ \((\(-2\) + 
                      x\^2)\)\ \((\(-1\) + x + x\^2)\)\^2\ \((\(-2\) + 2\ x + 
                      x\^2)\)\ \((2 - 4\ x - x\^2 + 
                      x\^3)\)\ \((\(-1\) + 7\ x - 5\ x\^2 - x\^3 + x\^4)\)\^2\ \((2 - 8\ x - 5\ x\^2 + 2\ x\^3 + 
                      x\^4)\)\ \((5 - 20\ x + 19\ x\^2 + 5\ x\^3 - 9\ x\^4 + x\^6)\)\^2\ \((\(-25\) + 15\ x + 164\ x\^2 - 17\ x\^3 - 216\ x\^4 - 17\ x\^5 + 92\ x\^6 + 9\ x\^7 - 16\ x\^8 - x\^9 + x\^10)\)\^2\ \((\(-1\) + 17\ x + 36\ x\^2 - 91\ x\^3 - 94\ x\^4 + 67\ x\^5 + 62\ x\^6 - 15\ x\^7 - 14\ x\^8 + x\^9 + x\^10)\)\^2, {"gS[3,{{2,20,0},{1,10,1,-.5,-19},{3,30,1,-.05},{4,20}},{{-31,32,3,3,10},{1,31,2,3,10},{2,3,2,2,10},{2,32,2,3,10},{2,5,2,2,7},{7,10,2,2,7},{16,19,2,2,2},{1,18,0,2,2},{3,17,0}}]"}}\)},{},
{  {"", 0, {}}},{},
{  {"", 0, {}}},{},
{  \({"86", \((3 + 
                      x)\)\ \((\(-79\) + 544\ x\^2 - 1156\ x\^4 + 1133\ x\^6 - 575\ x\^8 + 153\ x\^10 - 20\ x\^12 + x\^14)\)\^6, {{{13, \(-13\)}, 43}}}\)},{},
{  {"", 0, {}}},{},
{  \({"F90 Foster", \((\(-2\) + x)\)\ x\^2\ \((2 + 
                      x)\)\ \((3 + 
                      x)\)\ \((\(-3\) + 18\ x\^2 - 10\ x\^4 + x\^6)\)\^2\ \((5 - 140\ x\^2 + 331\ x\^4 - 266\ x\^6 + 96\ x\^8 - 16\ x\^10 + x\^12)\)\^2\ \((496 - 9344\ x\^2 + 62816\ x\^4 - 195312\ x\^6 + 330120\ x\^8 - 333884\ x\^10 + 214302\ x\^12 - 90198\ x\^14 + 25179\ x\^16 - 4612\ x\^18 + 532\ x\^20 - 35\ x\^22 + x\^24)\)\^2, {{{41, 21, 37, \(-37\), \(-21\), \(-41\)}, 15}}}\)},{},{},{},{},{},{},{},{},{},{  },{},
{  \({"Biggs-Smith", \((\(-2\) + x)\)\^18\ x\^17\ \((\(-4\) - x + x\^2)\)\^9\ \((\(-3\) + 3\ x\^2 + x\^3)\)\^16, {{{16, 
                      24, \(-38\), 17, 34, 48, \(-19\), 41, \(-35\), 
                      47, \(-20\), 34, \(-36\), 21, 14, 
                      48, \(-16\), \(-36\), \(-43\), 28, \(-17\), 21, 
                      29, \(-43\), 46, \(-24\), 
                      28, \(-38\), \(-14\), \(-50\), \(-45\), 21, 8, 
                      27, \(-21\), 20, \(-37\), 39, \(-34\), \(-44\), \(-8\), 
                      38, \(-21\), 25, 15, \(-34\), 18, \(-28\), \(-41\), 36, 
                      8, \(-29\), \(-21\), \(-48\), \(-28\), \(-20\), \(-47\), 14, \(-8\), \(-15\), \(-27\), 38, 24, \(-48\), \(-18\), 25, 38, 
                      31, \(-25\), 24, \(-46\), \(-14\), 28, 11, 21, 
                      35, \(-39\), 43, 36, \(-38\), 14, 50, 43, 
                      36, \(-11\), \(-36\), \(-24\), 45, 8, 19, \(-25\), 38, 
                      20, \(-24\), \(-14\), \(-21\), \(-8\), 
                      44, \(-31\), \(-38\), \(-28\), 37}, 1}}}\)},{},{},{},{},{},{},{},
{  },{},{},{},{},{},{},{},
{  },{},\({{"Balaban
11 cage, 64", 
                x\^12\ \((\(-6\) + x\^2)\)\^5\ \((\(-2\) + x\^2)\)\^12\ \((2 - 4\ x - x\^2 + x\^3)\)\^2\ \((\(-2\) - 6\ x + x\^2 + 
                      x\^3)\)\ \((4 + 4\ x - 6\ x\^2 - x\^3 + x\^4)\)\^4\ \((4 + 12\ x - 6\ x\^2 - 8\ x\^3 + x\^4 + x\^5)\)\^8, {{{44, 
                      26, \(-47\), \(-15\), 35, \(-39\), 11, \(-27\), 
                      38, \(-37\), 
43, 14, 28, 
                      51, \(-29\), \(-16\), 41, \(-11\), \(-26\), 
                      15, 
22, \(-51\), \(-35\), 36, 
                      52, \(-14\), \(-33\), \(-26\), \(-46\), 52, 26, 16, 43, 
                      33, \(-15\), 17, \(-53\), 
                      23, \(-42\), \(-35\), \(-28\), 30, \(-22\), 
                      45, \(-44\), 16, \(-38\), \(-16\), 50, \(-55\), 20, 
                      28, \(-17\), \(-43\), 47, 34, \(-26\), \(-41\), 
                      11, \(-36\), \(-23\), \(-16\), 41, 17, \(-51\), 
                      26, \(-33\), 47, 17, \(-11\), \(-20\), \(-30\), 21, 29, 
                      36, \(-43\), \(-52\), 10, 
                      39, \(-28\), \(-17\), \(-52\), 51, 26, 37, \(-17\), 
                      10, \(-10\), \(-45\), \(-34\), 17, \(-26\), 
                      27, \(-21\), 46, 53, \(-10\), 29, \(-50\), 35, 
                      15, \(-47\), \(-29\), \(-41\), 26, 33, 55, \(-17\), 
                      42, \(-26\), 
\(-36\), 16}, 1}}}}\),
{},{},{},{},{},{},{},{  },{},{},{},{},{},{  \({"Tutte
12 cage", 
                x\^28\ \((3 + 
                      x)\)\ \((\(-6\) + x\^2)\)\^21\ \((\(-2\) + x\^2)\)\^27, {{{17, 27, \(-13\), \(-59\), \(-35\), 35, 11, 13, \(-53\), 53, \(-27\), 21, 
                      57, 11, \(-21\), \(-57\), 59, \(-17\)}, 7}}}\)}};

4 regular

x =. ;graph4Data={{},{},{},{},
{  {"4,5,120", -(((1 + x))^4), {{{2, 3}, 3}}}},
{  {"AntiPrism3,48", 
                (x^3) (((2 + x))^2), {{{4, 2}, 3}, {{2, 3, 3, 2}, 2}}}},
{  \({"4,7,48", \(-\((\(-1\) + 
                        x)\)\)\ x\^2\ \((1 + x)\)\^2\ \((3 + x)\), {{{2, 0, 
                      3}, 7}}}, {"4,7,14,1", \(-\((\(-1\) - x + 2\ x\^2 + x\^3)\)\^2\), {{{2}, 7}, {{3}, 7}, {{3, 3, 2}, 5}, {{3, 2, 2}, 7}, {{2, 4}, 
                    7}}}\)},
{  \({"4,8,1152", 
                x\^6\ \((4 + x)\), {{{3}, 
                    8}, "gS[4,{{1,6,1},{.57,2,0,.571,-1}},{{1,4,1,1,3},{1,7,2,0,3},{2,8,2,0,3},{7,8}}]"}}, {"4,8,48", \((\(-2\) + 
                      x)\)\ x\^3\ \((2 + x)\)\^3, {{{3, 4, 6, 6, 2, 2, 4, 5}, 
                    1}}}, 
{"Antiprism4
16,1 16", 
                x\ \((2 + x)\)\^2\ \((\(-2\) + x\^2)\)\^2, {{{2}, 
                    8}, {{2, 0, 3}, 
                    8}, "gS[4,{{5,8,3}},{{2,2,-1}}]", "gS[4,{{.3,2,1,-1.57},{1,6}},{{0,0,0,4,2},1}]"}}, 
{"4,8,16,2", 
                x\^4\ \((2 + x)\)\ \((\(-4\) + 2\ x + x\^2)\), {{{4, 4, 2}, 
                    6}, {{\(-3\), 4, 2}, 7}, {{\(-2\), 3, 3}, 
                    8}, {{\(-3\), 3, 4}, 
                    8}}}, 
{"4,8,12", \((1 + 
                      x)\)\ \((\(-4\) + x + 
                      x\^2)\)\ \((\(-1\) + x + x\^2)\)\^2, {{{2, 4, 3, 4, 3, 
                      2}, 3}}}, 
{"4,8,4", 
                x\^2\ \((2 + x)\)\ \((\(-2\) + x\^2)\)\ \((\(-2\) + 2\ x + 
                      x\^2)\), {{{2, 3, 4}, 
                    7}, "gS[4,{{2,6},{.5,2,1,-1.05}},{{3,6},{1,5},{2,4}}]"}}\)},
{  \({"4,9,72", \(-\((\(-1\) + x)\)\^4\)\ \((2 + x)\)\^4, {{{4, \(-2\), 4}, 
                    3}}}, 
{"4,9,32,1", \((2 - 
                      x)\)\ \((\(-1\) + x)\)\ x\^2\ \((1 + x)\)\^2\ \((2 + 
                      x)\)\ \((3 + x)\), {{{4, \(-2\), \(-3\), 4, 
                      2, \(-2\), \(-4\), 3, 2, \(-4\)}, 
                    1}}}, 
{"4,9,18,1", \(-\((\(-1\) + x)\)\^2\)\ \((\(-1\) + 3\ x\^2 + x\^3)\)\^2, {{{3}, 
                    9}}}, 
{"4,9,18,2", \(-\((2 + x)\)\^2\)\ \((\(-1\) - 3\ x + x\^3)\)\^2, {{{2}, 9}, {{4}, 9}, {{\(-2\), 3}, 
                    9}}}, 
{"4,9,16,1", \((1 - 
                      x)\)\ x\ \((1 + x)\)\^2\ \((2 + x)\)\^2\ \((\(-4\) - 
                      x + x\^2)\), {{{\(-2\), 4, 3, \(-4\)}, 
                    5}}}, 
{"4,9,16,2", \(-\((\(-1\) + x)\)\^2\)\ x\^2\ \((1 + x)\)\ \((2 + x)\)\ \((\(-2\) + 3\ x + 
                      x\^2)\), {{{\(-3\), 4, 4, \(-3\)}, 
                    6}}}, 
{"4,9,12,1", \(-\((\(-1\) + x)\)\^3\)\ x\^2\ \((2 + x)\)\^2\ \((3 + x)\), {{{2, 3, 4}, 
                    9}}}, 
{"4,9,12,2", \((2 - 
                      x)\)\ \((\(-1\) + x)\)\^2\ \((1 + x)\)\^2\ \((2 + x)\)\^3, {{{2, 3, 2, 4, 2}, 
                    6}}}, 
{"4,9,8,1", \(-\((\(-1\) + x)\)\^2\)\ x\ \((1 + x)\)\ \((2 + x)\)\^2\ \((\(-4\) + x + x\^2)\), {{{2, 
                      3, \(-4\), \(-4\)}, 
                    5}}}, 
{"4,9,8,2", \((3 + x - 
                      x\^2)\)\ \((\(-1\) + x + x\^2)\)\^2\ \((1 + 3\ x + 
                      x\^2)\), {{{2, 3}, 
                    9}}}, 
{"4,9,8,3", \(-\((\(-1\) - 
                        x + x\^2)\)\)\ \((\(-1\) + x + x\^2)\)\^2\ \((\(-1\) + 3\ x + x\^2)\), {{{7, 4, 3, 3, 3, 0, 0, 0, 0, 3, 5, 6, 6, 4, 0, 0, 2}, 
                    1}}}, 
{"4,9,4,1", \(-x\)\ \((\(-1\) - x + 2\ x\^2 + x\^3)\)\ \((4 - 5\ x - 5\ x\^2 + 2\ x\^3 + 
                      x\^4)\), {{{7, 4, 3, 4, 4, 0, 0, 0, 0, 6, 3, 6, 3, 0, 
                      0, 3, 2}, 
                    1}}}, 
{"4,9,2,1", \(-\((\(-1\) + 
                        x)\)\)\ x\ \((1 + x)\)\ \((2 + x)\)\ \((4 - 6\ x - 
                      5\ x\^2 + 2\ x\^3 + x\^4)\), {{{\(-3\), 4}, 
                    9}, "gS[4,{{1,7},{.5,2,1,-.87}},{{3,4,3,0,2},1}]"}}, 
{"4,9,2,2", \(-\((2 + x)\)\)\ \((\(-1\) - 3\ x + 
                      x\^3)\)\ \((2 - 3\ x - 3\ x\^2 + 2\ x\^3 + 
                      x\^4)\), {{{7, 4, 3, 4, 4, 0, 3, 0, 0, 6, 3, 4, 5, 0, 
                      0, 0, 2}, 
                    1}}}, 
{"4,9,2,3", \((1 - 
                      x)\)\ \((\(-1\) + 3\ x\^2 + x\^3)\)\ \((1 - 5\ x - 
                      3\ x\^2 + 2\ x\^3 + x\^4)\), {{{2, \(-4\), 3, 
                      4, \(-3\)}, 6}}}\)},
{  \({"4,10,320", 
                x\^5\ \((\(-4\) + 2\ x + x\^2)\)\^2, {{{4}, 
                    10}, {{7, 5, 3, 4, 0, 0, 0, 0, 0, 0, 5, 3, 6, 6, 4, 5, 3, 
                      3}, 1}}}, 
{"4,10,240", \((\(-1\) + x)\)\^4\ \((1 + x)\)\^4\ \((4 + x)\), {{{3}, 10}, {{\(-3\), 5}, 
                    5}, {{7, 5, 3, 5, 0, 0, 0, 0, 0, 0, 3, 3, 5, 7, 5, 3, 3, 
                      3}, 1}}}, 
{"4,10,48", \((\(-1\) + x)\)\^2\ \((1 + x)\)\^5\ \((\(-8\) + x + x\^2)\), {{{7, 5, 2, 5, 0, 0, 0, 
                      2, 0, 0, 3, 3, 3, 7, 0, 3, 3, 3}, 1, 
                    7}}}, 
{"4,10,32", \((\(-2\) + 
                      x)\)\ x\^4\ \((2 + x)\)\^2\ \((\(-4\) + 2\ x + 
                      x\^2)\), {{{4, 5, 5, 0, 4, 2, 0, 0, 0, 0, 2, 2, 8, 6, 
                      6, 4, 2}, 1, 
                    1}}}, 
{"Antiprism5 20", 
                x\ \((1 + x)\)\^4\ \((\(-5\) + x\^2)\)\^2, {"gS[4,{{2,5},{1,5,1,-.63}}]", {{\(-2\), 2}, 5}, {{2, 3, \(-3\), \(-2\)}, 
                    5}}}, 
{"4,10,16,1", 
                x\^3\ \((2 + x)\)\^2\ \((\(-6\) + x\^2)\)\ \((\(-2\) + 
                      x\^2)\), {{{\(-4\), 4, 5}, 
                    5}, {{4, 5, 5, 0, 5, 2, 0, 0, 0, 0, 2, 2, 8, 6, 6, 3, 2}, 
                    1, 1}}}, 
{"4,10,16,2", 
                x\^2\ \((2 + x)\)\ \((x\^2 - 2)\)\^2\ \((\(-4\) + 2\ x + 
                      x\^2)\), {{{8, 5, 3, 0, 3, 0, 0, 0, 0, 0, 6, 2, 5, 6, 
                      5, 3, 4, 0, 2}, 
                    1}}}, 
{"4,10,16,3", \((1 + 
                      x)\)\ \((\(-4\) + x + 
                      x\^2)\)\ \((1 - 3\ x + x\^2 + x\^3)\)\^2, {{{8, 5, 3, 
                      0, 3, 0, 0, 0, 0, 0, 4, 2, 6, 4, 6, 4, 3, 0, 2}, 
                    1}}}, 
{"4,10,16,4", \((3 + 
                      x)\)\ \((\(-1\) + x\^2)\)\^3\ \((\(-4\) + x + 
                      x\^2)\), {{{2, 3, 4, 2, 4, 2, 3, 4, 2, 4}, 
                    1}, {{6, 5, 3, 0, 4, 0, 0, 0, 0, 0, 4, 2, 5, 6, 6, 3, 4, 
                      2}, 1}, {{2, 5, 5, 5, \(-3\), \(-2\), 3, \(-2\), 2, 5}, 
                    1}, "gS[4,{{1,9},{0,1,1,0,6}},{{4,6,-2,3,4,-3,0,-2},1}]", "gS[4,{{1,8},{.2,2,1,1.1,5}},{{4,6,4,4,0,-2},1}]"}}, 
{"4,10,16,5", \((\(-1\) + x)\)\^2\ \((1 + x)\)\^4\ \((3 + x)\)\ \((\(-4\) - x + x\^2)\), {{{7, 5, 7, 0, 2, 2, 0, 0, 0, 0, 2, 2, 
                      8, 6, 4, 3, 0, 3}, 1, 
                    9}}}, 
{"4,10,12", \((1 + x)\)\^2\ \((\(-1\) + x + x\^2)\)\^2\ \((4 - 8\ x + x\^3)\), {{{7, 5, 2, 0, 0, 2, 0, 0, 
                      0, 0, 4, 2, 7, 6, 6, 3, 2, 3}, 
                    1}}}, 
{"4,10,10", 
                x\ \((5 - 5\ x - 4\ x\^2 + 2\ x\^3 + x\^4)\)\^2, {{{4, 2}, 
                    5}, "gS[4,{{1,5},{.5,5,3,-π/5.}}]", {{8, 5, 3, 0, 
                      3, 0, 0, 0, 0, 0, 6, 2, 6, 4, 5, 4, 4, 0, 2}, 
                    1}}}, 
{"4,10,8,1", 
                x\^3\ \((2 - 4\ x + x\^3)\)\ \((\(-6\) + 4\ x\^2 + 
                      x\^3)\), {{{\(-4\), 5, 3, 4}, 
                    5, \(-1\)}, {{7, 5, 2, 0, 0, 3, 0, 0, 0, 0, 5, 2, 5, 5, 
                      5, 5, 3, 3}, 1}}}, 
{"4,10,8,2", 
                x\^3\ \((\(-2\) + x\^2)\)\ \((\(-4\) + 2\ x + 
                      x\^2)\)\ \((\(-2\) + 2\ x + x\^2)\), {{{4, 5, 3, 4}, 
                    5, \(-1\)}, {{\(-4\), 3, \(-4\), 4}, 
                    5, \(-2\)}}}, 
{"4,10,8,3", \((\(-1\) + x)\)\ \((1 + x)\)\^4\ \((8 - 9\ x - 9\ x\^2 + x\^3 + x\^4)\), {{{8, 5, 
                      2, 0, 0, 0, 0, 2, 0, 0, 7, 2, 3, 2, 4, 0, 3, 3, 2}, 
                    1}}}, 
{"4,10,8,4", 
                x\ \((1 + x)\)\^2\ \((\(-1\) + x + x\^2)\)\ \((16 - 4\ x - 
                      9\ x\^2 + x\^3 + x\^4)\), {{{7, 5, 2, 0, 0, 0, 0, 2, 0, 
                      0, 5, 2, 3, 5, 4, 5, 3, 3}, 
                    1}}}, 
{"4,10,8,5", 
                x\ \((1 + x)\)\ \((\(-1\) + x + x\^2)\)\ \((\(-4\) + 16\ x - 
                      9\ x\^2 - 8\ x\^3 + 2\ x\^4 + x\^5)\), {{{8, 5, 2, 0, 
                      0, 3, 0, 0, 0, 0, 5, 2, 5, 4, 5, 5, 3, 0, 2}, 
                    1}}}, 
{"4,10,8,7", \((\(-1\) + 
                      x)\)\ \((\(-4\) - x + 
                      x\^2)\)\ \((\(-1\) + x + 3\ x\^2 + x\^3)\)\^2, {{{4, 5, 
                      2, 0, 0, 3, 0, 0, 0, 0, 3, 2, 5, 7, 6, 4, 2, 2}, 
                    1}}}, 
{"4,10,8,8", 
                x\ \((1 + x)\)\^2\ \((2 + x)\)\ \((\(-1\) + x + 
                      x\^2)\)\ \((6 - 7\ x - x\^2 + x\^3)\), {{{4, 5, 2, 0, 
                      0, 2, 0, 0, 0, 0, 3, 2, 7, 7, 6, 3, 2, 2}, 
                    1}}}, 
{"4,10,4,1", 
                x\ \((\(-2\) + x\^2)\)\ \((2 - 4\ x + x\^3)\)\ \((\(-2\) + 
                      2\ x + 4\ x\^2 + x\^3)\), {{{4, 5, 3}, 
                    4}}}, 
{"4,10,4,2", 
                x\ \((1 + x)\)\ \((\(-1\) + x + x\^2)\)\ \((\(-4\) + 16\ x - 
                      5\ x\^2 - 8\ x\^3 + 2\ x\^4 + x\^5)\), {{{\(-4\), 5, 
                      3}, 7}}}, 
{"4,10,4,3", \((\(-1\) + x)\)\ \((1 + x)\)\^2\ \((\(-3\) + x\^2)\)\ \((\(-4\) + x + 
                      x\^2)\)\ \((\(-1\) + 2\ x + x\^2)\), {{{2, 3, 
                      5, \(-4\), 2, 4}, 
                    2}}}, 
{"4,10,4,4", \((\(-1\) + x)\)\^2\ x\ \((1 + x)\)\^3\ \((3 + x)\)\ \((\(-5\) + 
                      x\^2)\), {{{2, \(-3\), \(-3\), \(-2\), \(-4\)}, 2, 
                    3}}}, 
{"4,10,4,5", \((\(-1\) + x\^2)\)\^2\ \((\(-1\) + 2\ x + x\^2)\)\ \((\(-8\) - 5\ x + 2\ x\^2 + 
                      x\^3)\), {{{8, 5, 2, 0, 0, 0, 0, 0, 0, 0, 5, 2, 3, 4, 
                      4, 5, 3, 2, 2}, 
                    1}}}, 
{"4,10,4,6", \((\(-1\) + 
                      x)\)\ \((1 + x)\)\^3\ \((\(-1\) + 2\ x + 
                      x\^2)\)\ \((4 - 7\ x + x\^3)\), {{{7, 5, 2, 0, 0, 0, 0, 
                      2, 0, 0, 4, 2, 3, 5, 6, 3, 3, 3}, 
                    1}}}, 
{"4,10,4,7?",  (*\ 
                  isospecral\ non - 
                    isomorphicp6*) 
\((\(-1\) + 
                      x)\)\ \((1 + x)\)\^4\ \((\(-5\) + x\^2)\)\ \((\(-4\) + 
                      x + x\^2)\), {{{7, 5, 2, 0, 0, 0, 0, 2, 0, 0, 6, 2, 3, 
                      6, 4, 3, 4, 3}, 
                    1}, 
"gS[4,{{1,7},{.2,3,1,2.5,4}},{{2,2,0,2,2},1}]", "gS[4,{{1,6,1,.525},{.4,4,1,.785}},{{-7,10,1,-1,2},{-2,7,5,3,2},{3,9,2,0,2},{6,7,-4,3,2},{1,8,3,-1,2}}]", "gS[4,{{1,6,1,.525},{.4,4,1,.785}},{{-7,10,1,-1,2},{-1,7,4,3,2},{3,9,2,-2,2},{4,10,2,-3,2},{1,8,0,2,2}}]"}}, 
{"4,10,4,8", \((1 + x)\)\^4\ \((\(-3\) + x\^2)\)\ \((4 - 7\ x + x\^3)\), {{{7, 5, 2, 0, 0, 0, 0, 
                      2, 0, 0, 3, 2, 3, 7, 4, 3, 3, 3}, 
                    1}}}, 
{"4,10,4,9", \((2 + 
                      x)\)\ \((\(-2\) + x\^2)\)\ \((2 - 4\ x + 
                      x\^3)\)\ \((\(-2\) - 2\ x + 2\ x\^2 + x\^3)\), {{{8, 5, 
                      2, 0, 0, 2, 0, 0, 0, 0, 6, 2, 5, 5, 5, 4, 4, 0, 2}, 
                    1}}}, 
{"4,10,4,10", 
                x\ \((\(-2\) + 2\ x + x\^2)\)\ \((2 - 4\ x + 
                      x\^3)\)\ \((\(-2\) - 2\ x + 2\ x\^2 + x\^3)\), {{{8, 5, 
                      2, 0, 0, 2, 0, 0, 0, 0, 5, 2, 5, 5, 5, 5, 3, 0, 2}, 
                    1}}}, 
{"4,10,4,11", 
                x\^2\ \((2 + x)\)\ \((2 - 4\ x + x\^3)\)\ \((\(-6\) - 4\ x + 
                      2\ x\^2 + x\^3)\), {{{7, 5, 2, 0, 0, 3, 0, 0, 0, 0, 6, 
                      2, 5, 5, 5, 4, 4, 3}, 
                    1}}}, 
{"4,10,4,12", 
                x\^2\ \((2 + x)\)\ \((\(-2\) + x\^2)\)\ \((4 - 8\ x - 
                      6\ x\^2 + 2\ x\^3 + x\^4)\), {{{6, 5, 2, 0, 0, 3, 0, 0, 
                      0, 0, 3, 2, 6, 7, 3, 4, 4, 2}, 
                    1}}}, 
{"4,10,4,13", \((1 + x)\)\^4\ \((\(-1\) + 2\ x + x\^2)\)\ \((8 - 5\ x - 2\ x\^2 + x\^3)\), {{{3, 5, 2, 0, 
                      0, 2, 0, 0, 0, 0, 2, 2, 8, 7, 4, 3, 3, 2}, 
                    1}}}, 
{"4,10,4,14", 
                x\ \((\(-1\) + x\^2)\)\^2\ \((\(-5\) + 2\ x + 
                      x\^2)\)\ \((\(-1\) + 2\ x + x\^2)\), {{{8, 5, 3, 0, 3, 
                      0, 0, 0, 0, 0, 5, 2, 5, 5, 5, 5, 3, 0, 2}, 1, 
                    2}}}, 
{"4,10,4,15", \((\(-1\) + x)\)\^2\ \((1 + x)\)\^3\ \((4 - 11\ x - 5\ x\^2 + 3\ x\^3 + x\^4)\), {{{7, 5, 
                      3, 0, 4, 0, 0, 0, 0, 0, 3, 2, 5, 7, 5, 3, 3, 3}, 
                    1}}}, 
{"4,10,2,1", \((\(-1\) + 
                      x)\)\ \((1 + x)\)\ \((\(-1\) - 3\ x + x\^2 + 
                      x\^3)\)\ \((4 - 7\ x - 3\ x\^2 + 3\ x\^3 + 
                      x\^4)\), {{{2, \(-3\), 4, 4, 5, \(-4\), \(-2\), 
                      3, \(-3\), 4}, 
                    1}, {{5, \(-2\), \(-4\), \(-2\), \(-4\), 
                      2, \(-4\), \(-4\), \(-4\), \(-3\)}, 
                    1}}}, 
{"4,10,2,2", \((1 + x)\)\^2\ \((1 - 3\ x - x\^2 + x\^3)\)\ \((4 - 7\ x - 3\ x\^2 + 3\ x\^3 + 
                      x\^4)\), {{{\(-3\), 5, 4}, 
                    5}}}, 
{"4,10,2,3", \((1 + 
                      x)\)\ \((\(-1\) + x + x\^2)\)\ \((\(-4\) + 20\ x\^2 - 
                      9\ x\^3 - 8\ x\^4 + 2\ x\^5 + x\^6)\), {{{\(-4\), 3, 5, 
                      4}, 5, 1}}}, 
{"4,10,2,4", \((1 + x)\)\^2\ \((1 - 5\ x + x\^2 + x\^3)\)\ \((4 - 3\ x - 5\ x\^2 + x\^3 + 
                      x\^4)\), {{{5, \(-4\), 2, 3}, 5, 
                    2}}}, 
{"4,10,2,5", \((1 - 3\ x - 
                      2\ x\^2 + 2\ x\^3 + x\^4)\)\ \((4 + 9\ x - 7\ x\^2 - 
                      6\ x\^3 + 2\ x\^4 + x\^5)\), {{{\(-3\), 4, 2, \(-4\), 
                      2, 5, 2, \(-4\), 4, 2}, 
                    1}, "gS[4,{{1,7},{.7,3}},{{3,6,1,1,2},{1,4,0},{2,5,0}}]"}}, 
{"4,10,2,7", 
                x\ \((5 - 7\ x - 6\ x\^2 + 2\ x\^3 + x\^4)\)\ \((1 - 3\ x - 
                      2\ x\^2 + 2\ x\^3 + x\^4)\), {{{8, 5, 2, 0, 0, 0, 0, 0, 
                      0, 0, 7, 2, 3, 4, 5, 3, 3, 3, 2}, 
                    1}}}, 
{"4,10,2,8", 
                x\ \((1 + x)\)\^2\ \((3 - 5\ x - x\^2 + x\^3)\)\ \((\(-5\) - 
                      x + 3\ x\^2 + x\^3)\), {{{8, 5, 2, 0, 0, 0, 0, 2, 0, 0, 
                      7, 2, 3, 3, 4, 4, 0, 3, 2}, 
                    1}}}, 
{"4,10,2,9", \((\(-1\) + 
                      x)\)\ \((1 + x)\)\ \((\(-1\) + 2\ x + x\^2)\)\ \((4 + 
                      5\ x - 10\ x\^2 - 6\ x\^3 + 2\ x\^4 + x\^5)\), {{{8, 5, 
                      2, 0, 0, 0, 0, 0, 0, 0, 4, 2, 3, 4, 6, 3, 3, 2, 2}, 
                    1}}}, 
{"4,10,2,10", \((1 + x)\)\^2\ \((\(-1\) - 3\ x + x\^2 + x\^3)\)\ \((8 - 5\ x - 7\ x\^2 + x\^3 + 
                      x\^4)\), {{{7, 5, 2, 0, 0, 0, 0, 2, 0, 0, 6, 2, 3, 5, 
                      4, 4, 4, 3}, 
                    1}}}, 
{"4,10,2,11", \((\(-2\) + 
                      x)\)\ x\ \((1 + x)\)\^2\ \((2 + 
                      x)\)\ \((\(-3\) + x + x\^2)\)\^2, {{{8, 5, 2, 0, 0, 0, 
                      0, 0, 0, 0, 6, 2, 3, 5, 5, 2, 4, 2, 2}, 
                    1}}}, 
{"4,10,2,12", 
                x\^2\ \((1 + x)\)\ \((\(-1\) + x + x\^2)\)\ \((16 - 9\ x - 
                      8\ x\^2 + 2\ x\^3 + x\^4)\), {{{7, 5, 2, 0, 0, 4, 0, 0, 
                      0, 0, 5, 2, 5, 5, 4, 5, 3, 3}, 
                    1}}}, 
{"4,10,2,13", 
                x\ \((\(-3\) - 9\ x - 4\ x\^2 + 2\ x\^3 + x\^4)\)\ \((5 - 
                      5\ x - 4\ x\^2 + 2\ x\^3 + x\^4)\), {{{6, 5, 2, 0, 0, 
                      4, 0, 0, 0, 0, 5, 2, 5, 5, 4, 5, 4, 2}, 
                    1}}}, 
{"4,10,2,14", 
                x\ \((1 - 5\ x - 4\ x\^2 + 2\ x\^3 + x\^4)\)\ \((5 - 5\ x - 
                      4\ x\^2 + 2\ x\^3 + x\^4)\), {{{6, 5, 2, 0, 0, 3, 0, 0, 
                      0, 0, 5, 2, 5, 5, 5, 5, 4, 2}, 
                    1}}}, 
{"4,10,2,15", \((\(-1\) + 
                      x)\)\ x\ \((1 + x)\)\ \((\(-1\) + 2\ x + 
                      x\^2)\)\ \((5 - 6\ x - 6\ x\^2 + 2\ x\^3 + 
                      x\^4)\), {{{7, 5, 2, 0, 0, 3, 0, 0, 0, 0, 4, 2, 5, 5, 
                      6, 4, 3, 3}, 
                    1}}}, 
{"4,10,2,16", \((\(-1\) + 
                      x)\)\ x\ \((1 + x)\)\ \((1 - 5\ x + x\^2 + 
                      x\^3)\)\ \((\(-5\) - x + 3\ x\^2 + x\^3)\), {{{8, 5, 2, 
                      0, 0, 2, 0, 0, 0, 0, 5, 2, 5, 6, 4, 5, 3, 0, 2}, 
                    1}}}, 
{"4,10,2,17", 
                x\ \((1 + x)\)\^2\ \((\(-5\) + x\^2)\)\ \((\(-3\) + 
                      x\^2)\)\ \((\(-1\) + 2\ x + x\^2)\), {{{8, 5, 2, 0, 0, 
                      2, 0, 0, 0, 0, 3, 2, 6, 7, 2, 4, 0, 2, 2}, 
                    1}}}, 
{"4,10,2,18", 
                x\ \((5 - 5\ x - 4\ x\^2 + 2\ x\^3 + x\^4)\)\ \((1 - x - 
                      4\ x\^2 + 2\ x\^3 + x\^4)\), {{{8, 5, 3, 0, 3, 0, 0, 0, 
                      0, 0, 4, 2, 5, 5, 6, 4, 3, 0, 2}, 
                    1}}}, 
{"4,10,1,1", 
                x\^2\ \((1 + x)\)\ \((\(-3\) + x + x\^2)\)\ \((8 - 7\ x - 
                      6\ x\^2 + 2\ x\^3 + x\^4)\), {{{2, \(-4\), 4, \(-2\)}, 
                    4}, {{6, 5, 2, 0, 0, 3, 0, 0, 0, 0, 4, 2, 5, 5, 6, 4, 4, 
                      2}, 1}}}, 
{"4,10,1,2", \((\(-1\) + x)\)\ \((1 + x)\)\^2\ \((\(-4\) + 23\ x + 9\ x\^2 - 18\ x\^3 - 6\ x\^4 + 
                      3\ x\^5 + x\^6)\), {{{8, 5, 2, 0, 0, 0, 0, 0, 0, 0, 7, 
                      2, 3, 4, 4, 4, 3, 3, 2}, 
                    1}}}, 
{"4,10,1,3", 
                4 - 4\ x - 36\ x\^2 + 16\ x\^3 + 56\ x\^4 - 4\ x\^5 - 
                  28\ x\^6 - 4\ x\^7 + 4\ x\^8 + 
                  x\^9, {{{8, 5, 2, 0, 0, 0, 0, 0, 0, 0, 6, 2, 3, 4, 4, 4, 4, 
                      2, 2}, 1}}}, 
{"4,10,1,4", \((1 + x)\)\^2\ \((\(-1\) + x + x\^2)\)\ \((\(-4\) + 20\ x - 4\ x\^2 - 9\ x\^3 + 
                      x\^4 + x\^5)\), {{{8, 5, 2, 0, 0, 0, 0, 0, 0, 0, 7, 2, 
                      3, 4, 5, 4, 2, 3, 2}, 1}}}\)},
{  \({"4,11,22", \(-\((\(-1\) + 4\ x - 2\ x\^2 - 5\ x\^3 + 2\ x\^4 + x\^5)\)\^2\), {{{3}, 
                    11}}}, 
{"4,11,16,1", \(-\((\(-1\) + x)\)\)\ x\^2\ \((1 + x)\)\^2\ \((\(-2\) + 2\ x + x\^2)\)\ \((\(-10\) - 
                      8\ x + x\^2 + 
                      x\^3)\), {"gS[4,{{1,11}},{{9,2,2,0,0,0,0,3,0,0,0,7,4,3,3,4,0,3,4,2,2},1}]"}}, 
{"4,11,16,2", \(-x\)\ \((\(-3\) + x + 
                      x\^2)\)\ \((\(-1\) + x + x\^2)\)\^2\ \((\(-8\) - 7\ x + 
                      x\^2 + x\^3)\), {"gS[4,{{1,11}},{{9,2,3,0,0,0,0,3,0,0,0,7,3,4,5,2,4,0,4,2,2},1}]"}}, 
{"4,11,2", \(-x\)\ \((4 - x - 5\ x\^2 + 
                      x\^4)\)\ \((8 - 13\ x\^2 - x\^3 + 4\ x\^4 + 
                      x\^5)\), {"gS[4,{{1,7,1,-.675},{.15,4,1,-.77}},{{3,3,4},1}]", "gS[4,{{1,11,1,.428}},{{7,5,3,9,6,3,9,2,6,2,3},1}]"}}, 
{"4,11,1", \(-\((1 + x)\)\^2\)\ \((5 + 4\ x - 
                      48\ x\^2 + 26\ x\^3 + 38\ x\^4 - 16\ x\^5 - 11\ x\^6 + 
                      2\ x\^7 + 
                      x\^8)\), {"gS[4,{{1,9},{.4,2,1,-1.6}},{{3,4,2,0,3,0,2,-6},1}]"}}\)},
{  \({"4,12,768", \((\(-2\) + x)\)\^2\ x\^6\ \((2 + x)\)\^2\ \((4 + x)\), {{{5}, 12}}}\),
{"4,12,144", (-2 + 
                        x) (((-1 + x))^4) (((1 + x))^4) (2 + 
                        x) (4 + x), {{{3, 5}, 6}}},
{"4,12,64", (-2 + 
                        x) (x^5) (((2 + x))^2) (-8 - 8 x + 
                        2 (x^2) + (x^3)), {{{5, 5, 6, 7, 7}, 5}}},
\({"Cubocta
hedron 48", \((\(-2\) + x)\)\^3\ x\^3\ \((2 + x)\)\^5, {"gS[4,{{.7,4,1,.786},{1.5,4,0},{3,4,1,-.786}}]"}}\),
\({"4,12,48,1", \((\(-1\) + x)\)\^6\ x\ \((2 + x)\)\^2\ \((3 + x)\)\^2, {{{6, 3, \(-3\), 6}, 3}}}\),
{"4,12,48,2",(((-2 + x))^2) (((-1 + x))^2) x (((1 + x))^4) (((3 + x))^2),{"subGraph[{21,22,23,24},-1.,{{5,8},{6,11},{7,10},{9,12}},graphNo[4,24,1]]"}},{"4,12,32",(x^3) (((2 + x))^2) (-8 + 
                          (x^2)) (((-2 + (x^2)))^2),{"subGraph[{1,5,9,13},0.,{},graphNo[6,16,1]]"}},
\({"4,12,32,1", \((\(-2\) + x)\)\^2\ x\^4\ \((2 + x)\)\^3\ \((\(-4\) + 2\ x + x\^2)\), {{{5, \(-4\), 5, 
                        4, 5}, 8}}}\),
\({"Antiprism6 24", 
                  x\^3\ \((2 + x)\)\^4\ \((\(-2\) - 2\ x + x\^2)\)\^2, {"gS[4,{{2,6},{1,6,1,-.525}}]"}}\),
\({"4,12,24,2", \((\(-1\) + x)\)\^2\ x\^2\ \((1 + x)\)\^2\ \((4 + 
                        x)\)\ \((\(-3\) + x\^2)\)\^2, {{{\(-3\), 5, 
                        5, \(-3\)}, 6}}}\),
\({"4,12,24,3", \((\(-1\) + x)\)\^2\ \((1 + x)\)\^3\ \((\(-4\) + x + x\^2)\)\^3, {"gS[4,{{1,8,1,1.177},{2,4,1,.785}},{{4,4,4,4},1}]"}}\),
\({"4,12,24,4", \((\(-2\) + x)\)\^2\ x\^3\ \((2 + x)\)\^2\ \((\(-2\) + 2\ x + x\^2)\)\^2, {{{4, 4}, 
                      6}}}\),
{"4,12,16", 
                  (x^4) (-6 - 4 x + 2 (x^2) + (x^3)) (8 - 6 x - 
                        8 (x^2) + 2 (x^3) + (x^4)), {{{8, 6, 5, 5, 5, 5, 0, 
                        0, 0, 0, 0, 0, 5, 5, 7, 7, 7, 7, 5, 0, 4}, 1}}},
\({"4,12,12", \((\(-2\) + 
                        x)\)\ \((\(-1\) + x)\)\^2\ \((1 + x)\)\^2\ \((\(-4\) + 2\ x + x\^2)\)\ \((\(-1\) + 2\ x + x\^2)\)\^2, {{{4, 5, 3, \(-4\)}, 4}}}\),
{"4,12,8,1", (-1 + 
                        x) (x^2) (1 + x) (3 + 
                        x) (((-3 + (x^2)))^2) (-4 + x + 
                        (x^2)), {{{2, -3, 4, -4, -3, -2}, 
                      12}}},
{"4,12,8,2", 
                  (x^3) (2 + x) (2 - 2 x - 2 (x^2) + 
                        (x^3)) (-8 - 14 x - 2 (x^2) + 4 (x^3) + 
                        (x^4)), {{{9, 6, 6, 5, 5, 5, 0, 0, 0, 0, 0, 0, 5, 5, 
                        5, 7, 7, 7, 5, 0, 0, 3}, 1}}},\({"Chvatal 8", \((\(-1\) + x)\)\^4\ x\^2\ \((1 + 
                        x)\)\ \((3 + x)\)\^2\ \((\(-4\) + x + 
                        x\^2)\), {"gS[4,{{0.2,2,0},{.6,5,0,-.315},{1.,5.,-1,.3 3.1416}},{{-1,6,1,1,2},{-3,9,1,1,4},{-7,8},{1,6,1},{2,3,1},{3,5,1},{5,5,1},{6,1,1}}]", 
{{5, \(-3\), \(-3\), \(-3\), 3, 3, 3, 6, 6, 6, 
                        6, \(-5\)}, 1, 3}}}\),
{"4,12,4,1", 
                  (x^2) (((1 + x))^2) (-5 + (x^2)) (4 + 3 x - 
                        14 (x^2) - 8 (x^3) + 2 (x^4) + (x^5)), {{{5, 6, 6, 
                        5, 5, 5, 6, 0, 0, 0, 0, 0, 6, 5, 5, 7, 7, 7, 0, 0, 0, 
                        2}, 1}}},
{"4,12,4,2", 
                  (x^3) (-4 - x + (x^2)) (8 + 11 x - 17 (x^2) - 
                        21 (x^3) + (x^4) + 5 (x^5) + (x^6)), {{{8, 6, 5, 5, 
                        5, 6, 6, 0, 0, 0, 0, 0, 6, 5, 7, 7, 7, 5, 0, 0, 4}, 
                      1}}},{"4,12,4,3",(-1 + x) x (1 + x) (-3 + 
                          (x^2)) (((-4 + x + (x^2)))^2) (-1 + 
                          2 x + (x^2)),{"subGraph[{21,22,23,24},-1.,{{5,9},{6,10},{7,12},{8,11}},graphNo[4,24,1]]"}},\({"4,12,2", \((1 - 2\ x - 9\ x\^2 - 4\ x\^3 + 
                        2\ x\^4 + x\^5)\)\ \((4 + 13\ x + 2\ x\^2 - 
                        17\ x\^3 - 8\ x\^4 + 2\ x\^5 + 
                        x\^6)\), {"gS[4,{{1.,7},{.6,4,0,-.5},{0,1}},{{5,2,0}}]"}}\),
{"4,12,1,1", 
                  x (12 + 24 x - 60 (x^2) - 108 (x^3) + 68 (x^4) + 
                        132 (x^5) - 44 (x^7) - 8 (x^8) + 4 (x^9) + 
                        (x^10)), {{{5, 6, 5, 6, 5, 5, 6, 0, 0, 0, 0, 0, 6, 
                        5, 8, 5, 7, 7, 0, 0, 3}, 1}}},
{"4,12,1,2", 
                  x (1 + x) (12 - 28 x - 120 (x^2) - 4 (x^3) + 
                        121 (x^4) + 31 (x^5) - 35 (x^6) - 11 (x^7) + 
                        3 (x^8) + (x^9)), {{{9, 6, 6, 5, 5, 6, 0, 0, 0, 0, 
                        0, 0, 4, 5, 5, 7, 8, 5, 5, 0, 0, 3}, 1}}}},
{  \({"4,13,52", \(-\((1 - 4\ x + x\^2 + x\^3)\)\^4\), {{{5, \(-5\), 5, \(-5\)}, 
                    6}}}, {"4,13,2", \(-\((\(-1\) + x)\)\)\ \((5 - 6\ x - 
                      16\ x\^2 - 4\ x\^3 + 3\ x\^4 + x\^5)\)\ \((\(-1\) + 
                      3\ x + 6\ x\^2 - 12\ x\^3 - 7\ x\^4 + 2\ x\^5 + 
                      x\^6)\), {{{9, 2, 8, 7, 7, 6, 6, 5, 5, 4, 0, 0, 0, 8, 
                      7, 3, 0, 3, 0, 3}, 1}}}\)},{  \({"4,14 336", \((4 + 
                      x)\)\ \((\(-2\) + x\^2)\)\^6, {"gS[4,{{1,14}},{{3,5},7}]"}}, {"4,14 96", \((\(-1\) + 
                      x)\)\ x\^2\ \((1 + x)\)\^4\ \((\(-64\) + 36\ x + 
                      64\ x\^2 - 17\ x\^3 - 17\ x\^4 + x\^5 + 
                      x\^6)\), {"gS[4,{{1,14,1,-2.9}},{{7,7,5,2,10,11,11,10,10,3,3,12,12,12,7,5},1}]"}}, {"4,14,48,1", 
                x\^2\ \((1 + x)\)\^4\ \((16 - 100\ x + 16\ x\^2 + 85\ x\^3 - 
                      4\ x\^4 - 18\ x\^5 + 
                      x\^7)\), "gS[4,{{1,14}},{{11,2,2,0,0,0,2,3,0,0,0,2,0,0,8,3,3,2,0,0,3,5,6,3,3,3},1}]"}, {"4,14,48,2", \((1 + x)\)\^4\ \((\(-1\) - 2\ x + x\^2 + 
                      x\^3)\)\ \((\(-44\) + 11\ x + 60\ x\^2 + 4\ x\^3 - 
                      15\ x\^4 - x\^5 + 
                      x\^6)\), "gS[4,{{1,14}},{{8,2,2,0,0,0,2,2,0,0,2,0,0,0,7,3,3,2,0,0,5,7,6,3,3,2},1}]"}, {"4,14,48,3", \((\(-1\) + x)\)\ x\ \((1 + x)\)\^4\ \((2 + 
                      x)\)\ \((\(-48\) + 4\ x + 58\ x\^2 + 5\ x\^3 - 
                      15\ x\^4 - x\^5 + 
                      x\^6)\), "gS[4,{{1,14,1,-2.92}},{{7,7,4,2,10,11,11,8,10,3,3,12,12,12},1}]"}, {"4,14,48,4", \((\(-1\) + x)\)\ x\^3\ \((1 + x)\)\^3\ \((\(-5\) + 2\ x + 
                      x\^2)\)\ \((8 - 8\ x - 11\ x\^2 + 
                      x\^4)\), "gS[4,{{1,14,1,-2.92}},{{6,6,4,4,4,10,11,10,8,3,3,12,12,12,4,0,0,2},1}]"}, {"Antiprism7 28", 
                x\ \((\(-7\) - 7\ x + x\^3)\)\^2\ \((\(-1\) - x + 2\ x\^2 + x\^3)\)\^2, {"gS[4,{{2,7},{1,7,1,-.45}}]"}}, {"4,14,28,2", \((4 + 
                      x)\)\ \((1 - x - 2\ x\^2 + x\^3)\)\^2\ \((\(-1\) - x + 2\ x\^2 + x\^3)\)\^2, {"gS[4,{{1,14,1,.675}},{{3,-5},7}]"}}, {"14,4 2", x\ \((1 - x - 2\ x\^2 + x\^3)\)\ \((\(-1\) - x + 2\ x\^2 + 
                      x\^3)\)\ \((7 + 20\ x - 15\ x\^2 - 32\ x\^3 - 6\ x\^4 + 
                      4\ x\^5 + 
                      x\^6)\), {"gS[4,{{1,14,1,.675}},{{3,-2,3},10}]"}}\)},
{  \({"4,15,20", \(-\((\(-2\) + 
                        x)\)\)\ \((2 + x)\)\^5\ \((1 - 3\ x + x\^2)\)\^2\ \((\(-1\) + x + x\^2)\)\^2, {"gS[4,{{1.,5,0,-Pi/10.},{.3,5,1,-3Pi/10.},{.6,-5,1,-3Pi/10.}}]"}}, {"4,15,10", \(-\((\(-2\) + x)\)\)\ \((2 + 
                      x)\)\ \((1 + 7\ x + 8\ x\^2 - 11\ x\^3 - 7\ x\^4 + 2\ x\^5 + x\^6)\)\^2, {"gS[4,{{1.,5,0,-Pi/10.},{.3,5,2,-3Pi/10.},{.6,-5,1,-3Pi/10.}}]"}}\)},
{  \({\*""4,(Subscript[C, 4])\[Cross](Subscript[C, 4]),384"", \((\(-2\) + x)\)\^4\ x\^6\ \((2 + x)\)\^4\ \((4 + 
                      
                      x)\), {"gS[4,{{2,8,1,.385},{1,8,1,.385+2Pi,-1}},{{1,6,2,2,2},{2,5,2,2,2},{9,14,2,2,2},{10,13,2,2,2},{1,9,1,1,8}}]"}}, 
{"4,16,32", \((\(-2\) + x)\)\^3\ x\^3\ \((2 + 
                      x)\)\ \((\(-2\) + x\^2)\)\^2\ \((2 + 4\ x + x\^2)\)\^2, {"gS[4,{{2,8,1,.385},{1,8,1,2.Pi,-1}},{{1,5,1,1,4},{9,13,1,1,4},{1,9,1,1,8}}]"}}, {"Antiprism8 32", 
                x\ \((2 + x)\)\^2\ \((\(-2\) + x\^2)\)\^2\ \((\(-2\) - 8\ x - 8\ x\^2 + x\^4)\)\^2, {"gS[4,{{2,8},{1,8,1,-.4}}]"}}, {"4,16,16,1", \((\(-2\) + x)\)\ \((2 + x)\)\ \((4 + 
                      x)\)\ \((\(-2\) + x\^2)\)\^2\ \((2 - 4\ x\^2 + x\^4)\)\^2, {{{5, 7, 9, 11}, 
                    4}}}, {"4,16,16,2", \((\(-2\) + 
                      x)\)\ x\^4\ \((2 + x)\)\^4\ \((\(-2\) - 2\ x + x\^2)\)\^2\ \((\(-4\) + 2\ x + 
                      x\^2)\), {"gS[4,{{.54,4,1,-.78},{1,8,1,-.4},{.3,4,1,2.35}}]"}}\)},{},
{  \({"4,18,1296", \((\(-1\) + x)\)\^6\ \((1 + x)\)\^6\ \((4 + x)\)\ \((\(-7\) + x\^2)\)\^2, {{{7, 5}, 
                    9}}}, {"4,18,36,1", \((\(-2\) + x)\)\^2\ \((2 + x)\)\^2\ \((4 + 
                      x)\)\ \((\(-1\) - 3\ x + x\^3)\)\^2\ \((1 - 3\ x + x\^3)\)\^2, {"gS[4,{{1.,18}},{{-7,5},9}]"}}, {"4,18,36,2", \((\(-1\) + x)\)\^2\ \((1 + x)\)\^2\ \((4 + 
                      x)\)\ \((1 - 3\ x\^2 + x\^3)\)\^2\ \((\(-1\) + 3\ x\^2 + x\^3)\)\^2, {"gS[4,{{1.,18}},{{-7,9},9}]"}}, {"4,18,24", \((\(-3\) + x)\)\ \((\(-2\) + x)\)\^2\ \((\(-1\) + x)\)\^3\ x\^4\ \((1 + x)\)\^3\ \((2 + x)\)\^2\ \((3 + x)\)\ \((4 + x)\), {{{7, 5, 3}, 
                    6}}}, {"4,18,18,1", \((\(-1\) + x)\)\^2\ \((1 + x)\)\^2\ \((4 + 
                      x)\)\ \((\(-19\) + 24\ x\^2 - 9\ x\^4 + x\^6)\)\^2, {"gS[4,{{1.,18}},{{-7,3,-3,7},7}]"}}, {"4,18,18,2", 
                x\ \((\(-3\) - x + x\^2)\)\^2\ \((\(-3\) + 9\ x + 3\ x\^2 - 16\ x\^3 - 6\ x\^4 + 3\ x\^5 + x\^6)\)\^2, {"gS[4,{{1.,18}},{{-6,8},9}]"}}, {"4,18,18,3", 
                x\^3\ \((2 + x)\)\^2\ \((9 - 45\ x + 36\ x\^2 + 8\ x\^3 - 12\ x\^4 + x\^6)\)\^2, {"gS[4,{{1.,18}},{{-4,8},9}]"}}, {"Antiprism9", 
                x\^3\ \((2 + x)\)\^2\ \((\(-9\) - 9\ x + x\^3)\)\^2\ \((\(-1\) - 3\ x + x\^3)\)\^2, {"gS[4,{{2,9},{1,9,1,-.36}}]"}}, {"4,18,2", 
                0, {"gS[4,{{1.,6,0},{.2885,6,2,-Pi/6.},{.577,-6,1,-Pi/6.}}]"}}\)},
{  \({"Robertson 24", \(-\((\(-1\) + x)\)\^2\)\ \((\(-3\) + x\^2)\)\^2\ \((\(-5\) + x + 
                      x\^2)\)\ \((\(-4\) + x + x\^2)\)\^2\ \((\(-3\) + x + x\^2)\)\^2\ \((\(-1\) + x + 
                      x\^2)\), {"gS[4,{{1,19}},
{{8,4,7,4,8,5,7,4,7,8,4,5,7,8,4,8,4,8,4},1 }]"}}\)},
\({ (*20*) {"Folkman 3840", 
                x\ \((1 + x)\)\^4\ \((2 + x)\)\^2\ \((\(-5\) + x\^2)\)\^2\ \((1 + 2\ x - 6\ x\^2 - 2\ x\^3 + x\^4)\)\^2, {"gS[4,{{1,20}},{{-2,-4,-6,-8,10,8,6,4,2,10,-3,-5,-7,-9,9,7,5,3,-9,9},1}]", "gS[4,{{2,10},{1,10,1,-.32}}]"}}, 
{"4,20,20", \((\(-2\) + x)\)\ x\ \((2 + 
                      x)\)\ \((1 - 5\ x - 8\ x\^2 + x\^4)\)\^2\ \((5 - 5\ x - 4\ x\^2 + 2\ x\^3 + x\^4)\)\^2, {"gS[4,{{.54,5,1,-.3},{1,10,1},{.3,5,1,2.85}}]"}}}\),{  \({"4,21,14", \((2 - x)\)\ \((2 + 
                      x)\)\ \((1 + 22\ x - 57\ x\^2 - 63\ x\^3 + 73\ x\^4 + 46\ x\^5 - 25\ x\^6 - 13\ x\^7 + 2\ x\^8 + x\^9)\)\^2, {"gS[4,{{1.,7,0},{.35,7,2,-.45},{.69,-7,1,-.44}}]"}}\)},{},{},
{  \({"SmallRhombi
cuboctahedron", \((\(-3\) + x)\)\^3\ \((\(-1\) + x)\)\^2\ x\^4\ \((1 + x)\)\^6\ \((3 + x)\)\^2\ \((\(-4\) + x + x\^2)\)\^3, {"gS[4,{{5,4,1,.78},{3,8,1,1.175},{2,8,1,-1.96},{1,4,1, .78}}]"}}, 
{"4,24,96", \((\(-2\) + x)\)\^7\ x\^3\ \((2 + x)\)\^5\ \((\(-2\) + 2\ x + x\^2)\)\^4, {"gS[4,{{1,24}},{{1,5,3,3,8},{1,15,3,3,8},{2,9,4,4,8},{3,20,4,4,8},{5,12,4,4,8},{6,23,4,4,8}}]"}}\)},{},
\({ (*26*) {"cage4_6
32", \((4 + 
                      x)\)\ \((\(-3\) + x\^2)\)\^12, {"gS[4,{{1,26}},{{7,-11},13}]"}}}\),\({ (*27*) {"Doyle", \(-\((\(-1\) + x)\)\^4\)\ \((2 + x)\)\^4\ \((2 - 6\ x + x\^3)\)\^6, {"gS[4,{{2,9,2,-1.2},{1,9,4,-.1},{3,9,1,.177}},{{-10,19,1,1,9},{1,10,-9},{1,21,-20}}]", "gS[4,{{1,27}},{{13,-11,10,13,7,10,4,7,-8},6}]"}}, 
{"4,27,18", \(-\((\(-2\) + 
                        x)\)\)\ \((\(-1\) + x)\)\^2\ \((1 + x)\)\^2\ \((2 + x)\)\^3\ \((1 + 75\ x + 27\ x\^2 - 190\ x\^3 - 3\ x\^4 + 99\ x\^5 - x\^6 - 18\ x\^7 + x\^9)\)\^2, {"gS[4,{{1.,9,0},{.67,9,2,-.32},{.82,-9,1,-.33}}]"}}}\),
\({ (*28*) {"4,28,28", \((\(-2\) + 
                      x)\)\ \((2 + x)\)\ \((4 + 
                      x)\)\ \((\(-2\) + x\^2)\)\^6\ \((\(-8\) + 20\ x\^2 - 12\ x\^4 + x\^6)\)\^2, {{{9, 11, \(-11\), \(-9\)}, 28}}}}\),{},
\({ (*30*) {"Icosido
decahedron", \((\(-2\) + x)\)\ \((2 + x)\)\^2\ \((\(-4\) - 2\ x + 
                      x\^2)\)\ \((1 - 4\ x - 4\ x\^2 + x\^3 + x\^4)\)\^2\ \((\(-11\) - 78\ x - 29\ x\^2 + 95\ x\^3 + 56\ x\^4 - 21\ x\^5 - 15\ x\^6 + x\^7 + x\^8)\)\^2, {"gS[4,{{7,5,1,.94},{4.5,5,0,.31,-1},{3.2,10,1,.65,-1},{2.,5,0,.3,-1},{1,5,1,.85,-1}},{{21,26,1,1,5},{21,27,1,1,4},{11,21,2,1,5},{12,21,2,1,5},{1,6,1,1,5},{2,6,1,1,4},{6,11,1,2,5},{6,12,1,2,5},{25,1,0},{1,9,0}}]"}}, {"4,30,", \((\(-2\) + x)\)\^11\ x\^5\ \((1 + x)\)\^4\ \((2 + x)\)\^5\ \((3 + x)\)\^4, {"gS[4,{{1,30}},{{1,15,10,10,3},{1,24,10,10,3},{2,8,10,10,3},{2,13,10,10,3},{3,17,10,10,3},{3,22,10,10,3},{4,30,10,10,3},{4,11,10,10,3},{5,21,10,10,3},{5,26,10,10,3},{6,15,10,10,3},{6,18,10,10,3},{7,23,10,10,3},{7,29,10,10,3},{8,26,10,10,3},{9,20,10,10,3}}]"}}}\),{},\({ (*32*) {"4,32 ", \((\(-2\) + x)\)\ x\^4\ \((2 + 
                      x)\)\ \((\(-2\) + x\^2)\)\ \((2 - 4\ x + 
                      x\^2)\)\ \((\(-2\) + 2\ x + x\^2)\)\ \((2 + 4\ x + 
                      x\^2)\)\ \((\(-2\) - 4\ x + x\^3)\)\^2\ \((\(-4\) - 
                      4\ x + 2\ x\^2 + 
                      x\^3)\)\ \((8 - 2\ x - 10\ x\^2 + x\^4)\)\^2, {"gS[4,{{7,4,1,.79},{4.5,8,1,1.2},{3.2,8,1,-1.9},{2,8,1,1.2},
{1,4,1,3.92}}]"}}}\),{  \({"4,33, ", \(-\((\(-2\) + x)\)\)\ \((2 + 
                      x)\)\ \((1 + 183\ x + 1054\ x\^2 + 230\ x\^3 - 3319\ x\^4 - 1481\ x\^5 + 3768\ x\^6 + 1721\ x\^7 - 1943\ x\^8 - 882\ x\^9 + 480\ x\^10 + 220\ x\^11 - 53\ x\^12 - 25\ x\^13 + 2\ x\^14 + x\^15)\)\^2, {"gS[4,{{1.,11,0},{.79,11,2,-Pi/11.},{.89,-11,1,-Pi/11.}}]"}}\)},{},{  {"Odd (Subscript[O, 4])",-(((-2 + x))^14) (((1 + x))^14) (((3 + x))^6),{"subGraph[{2,4,10,13,16,18,23,25,29,33,35,39,43,46,48},0.,{},graphNo[7,50,1]]"}}},{},{},{},{},
{  },{},{},{},{},{},{},{},{},{},
{  },{},{},{},{},{},{},{},{},{},
\({ (*60*) {"SmallRhombicosi
dodecahedron", x\ \((6 + 21\ x - 6\ x\^2 - 13\ x\^3 + x\^5)\)\ \((\(-2\) + 
                      9\ x - 2\ x\^2 - 9\ x\^3 + 
                      x\^5)\)\ \((220 + 3400\ x + 14344\ x\^2 - 1860\ x\^3 - 116109\ x\^4 - 123120\ x\^5 + 319268\ x\^6 + 510730\ x\^7 - 363760\ x\^8 - 856626\ x\^9 + 130056\ x\^10 + 743666\ x\^11 + 74685\ x\^12 - 364602\ x\^13 - 85822\ x\^14 + 105202\ x\^15 + 33370\ x\^16 - 18040\ x\^17 - 6734\ x\^18 + 1790\ x\^19 + 747\ x\^20 - 94\ x\^21 - 43\ x\^22 + 2\ x\^23 + x\^24)\)\^2, {"gS[4,{{5,5,1},{3.8,10,1,.3},{3,15,1,3.75},{2,15,1,3.55},{1,10,1,3.7},{.5,5,1,.2}}]"}}}\),{},{},{},{},{},{},{},{},{},
\({ (*70*) {"Meredith", \((\(-1\) + x)\)\^10\ x\^21\ \((1 + x)\)\^11\ \((3 + x)\)\ \((\(-13\) + 
                      x\^2)\)\ \((\(-45\) - 39\ x + 169\ x\^2 + 3\ x\^3 - 26\ x\^4 + x\^6)\)\^4, {"gS[4,{{10,15,1,.25},{8,20,1,0,-1},{6,20,1,0,-1},{3,15,1,0.2,-1}},{{-16,17,2,2,10},{-36,37,2,2,10},{-58,59,3,3,4},{-56,70},{1,16,3,4,5},{1,17,3,4,5},{2,18,3,4,5},{2,35},{3,16,3,4,5},{3,17,3,4,5},{5,19,3,4,5},{16,18,4,4,5},{19,21,4,4,4},{18,38,4,4,5},{19,39,4,4,5},{37,58,4,3,5},{36,58,4,3,5},{37,55},{55,57},{36,38,4,4,5},{39,41,4,4,4},{36,58,4,3,5},{37,58,4,3,5},{37,56,4,3,5},{36,56,4,3,5},{38,57,4,3,5},{39,60,4,3,5},{56,67},{58,62},{59,70},{61,65},{64,68}}]"}}, {"xMeredith", \((\(-1\) + x)\)\^10\ x\^21\ \((1 + x)\)\^11\ \((3 + 
                      x)\)\ \((\(-13\) + 
                      x\^2)\)\ \((9 + 117\ x + 169\ x\^2 - 9\ x\^3 - 26\ x\^4 + x\^6)\)\^2\ \((45 - 195\ x + 169\ x\^2 + 15\ x\^3 - 26\ x\^4 + x\^6)\)\^2, {"gS[4,{{10,15,1,.25},{8,20,1,0,-1},{6,20,1,0,-1},{3,15,1,0.2,-1}},{{-16,17,2,2,10},{-36,37,2,2,10},{1,16,3,4,5},{1,17,3,4,5},{2,18,3,4,5},{2,35},{3,16,3,4,5},{3,17,3,4,5},{5,19,3,4,5},{16,18,4,4,5},{19,21,4,4,4},{18,38,4,4,5},{19,39,4,4,5},{37,58,4,3,5},{36,58,4,3,5},{36,70},{37,55},{55,57},{36,38,4,4,5},{39,41,4,4,4},{36,58,4,3,5},{37,58,4,3,5},{37,56,4,3,5},{36,56,4,3,5},{38,57,4,3,5},{39,60,4,3,5}}]"}}}\),{},{},{},{},{},{},{},{},{},
{  },{},{},{},{},{},{},{},{},{},
{  },{},{},{},{},{},{},{},{},{},
{},{},{},{},{},{},{},{},{},{}};

5 regular

x =. ;graph5Data={{},{},{},{},{},
{  {"5,6,720", (((1 + x))^5), {{{3, -2, 2, 3}, 
                    3}, "gS[5,{{1,5,3},{0,1}},{{1,1,1}}]"}}},{},{  \({"5,8,128", \((\(-1\) + x)\)\^2\ \((1 + x)\)\^4\ \((3 + x)\), {"gS[5,{{1.,8}},{{1,2,1},{1,3,3}}]", 
"gS[5,{{1.,8}},{{1,4,1},{1,3,0}}]", "gS[5,{{1.,8}},{{1,3,0},{1,4,2},{2,2,2}}]"}}, 
{"5,8,16", \((\(-1\) + 
                      x)\)\ \((1 + x)\)\^2\ \((\(-1\) + 2\ x + x\^2)\)\^2, {{{\(-2\), 4, 2}, 
                    4}, "gS[5,{{2,6},{.8,2}},{{1,2,2},{2,3,2},{4,2,2}}]", "gS[5,{{.3,2,1,-1.57},{1,6}},{{0,0,3,4,2,0,4,-3},1}]"}}\)},{},
{  \({"5,10,320", \((1 + x)\)\^5\ \((\(-5\) + x\^2)\)\^2, {{{1, 2, 5}, {1, 3, 
                      9}}, "gS[5,{{.5,5,1,.95},{1,5,1,-.315}}]"}}, 
{"5,10,32", \((\(-1\) + x)\)\^2\ \((1 + x)\)\^4\ \((3 + x)\)\ \((\(-5\) + x\^2)\), {{{3, 5, 5, 4, 4, 0, 0, 0, 0, 0, 6, 4, 4, 6, 6, 
                      4, 0, 0, 0, 0, 4, 6, 6, 7, 5, 3, 4}, 
                    1}}}, 
{"5,10,20", 
                x\^4\ \((1 + x)\)\ \((\(-4\) + 2\ x + x\^2)\)\^2, {{{\(-2\), 
                      5, 5, 2}, 
                    5}}}, 
{"5,10,16,1", \((\(-1\) + x)\)\^2\ \((1 + x)\)\^3\ \((\(-5\) + 2\ x + x\^2)\)\ \((\(-1\) + 2\ x + 
                      x\^2)\), {{{7, 4, 4, 5, 4, 0, 0, 0, 0, 0, 6, 6, 6, 4, 
                      6, 4, 4, 0, 0, 0, 4, 3, 3, 6, 0, 0, 3, 3}, 
                    1}}}, 
{"5,10,16,2", \((\(-2\) + 
                      x)\)\ x\^4\ \((2 + x)\)\^2\ \((\(-2\) + 3\ x + 
                      x\^2)\), {{{3, 4, 6, 4, 0, 0, 4, 0, 0, 0, 6, 3, 3, 3, 
                      3, 0, 0, 0, 0, 0, 7, 7, 7, 7, 5, 3, 3, 3}, 
                    1}}}, 
{"5,10,10", \((1 + 
                      x)\)\ \((5 - 5\ x - 4\ x\^2 + 2\ x\^3 + x\^4)\)\^2, {"gS[5,{{1,5},{.7,5,3,-π/5.}}]"}}, 
{"5,10,8", 
                x\^2\ \((2 + x)\)\ \((\(-2\) + x\^2)\)\^2\ \((\(-2\) + 3\ x + 
                      x\^2)\), {{{8, 4, 4, 4, 0, 0, 0, 0, 0, 0, 7, 3, 3, 3, 
                      3, 0, 0, 0, 0, 0, 3, 7, 7, 7, 5, 3, 3, 3, 2}, 
                    1}}}, 
{"5,10,4,1", \((\(-1\) + 
                      x)\)\ \((1 + x)\)\^2\ \((\(-1\) + 2\ x + 
                      x\^2)\)\ \((5 - 6\ x - 6\ x\^2 + 2\ x\^3 + 
                      x\^4)\), {{{8, 4, 3, 6, 0, 0, 0, 0, 0, 0, 7, 3, 2, 3, 
                      3, 4, 3, 3, 0, 0, 3, 7, 5, 7, 0, 0, 2, 0, 2}, 
                    1}}}, 
{"5,10,4,2", 
                x\^2\ \((1 + x)\)\ \((2 + x)\)\^2\ \((\(-6\) + 
                      x\^2)\)\ \((\(-2\) + x\^2)\), {{{7, 3, 3, 7, 0, 0, 3, 
                      0, 0, 0, 3, 7, 7, 0, 0, 0, 0, 2, 0, 0, 8, 2, 2, 2, 2, 
                      2, 2, 3, 2}, 1}}}, 
{"5,10,2,1", 
                x\^2\ \((\(-6\) - 4\ x + 2\ x\^2 + x\^3)\)\ \((2 - 6\ x - 
                      2\ x\^2 + 3\ x\^3 + x\^4)\), {{{4, 3, 4, 4, 0, 0, 0, 0, 
                      0, 0, 7, 7, 6, 3, 3, 3, 0, 0, 0, 0, 3, 8, 3, 7, 6, 4, 
                      3, 3}, 1}}}, 
{"5,10,2,2", 
                x\ \((\(-1\) + x + x\^2)\)\ \((16 + 31\ x - 6\ x\^2 - 
                      23\ x\^3 - 3\ x\^4 + 4\ x\^5 + x\^6)\), {{{4, 3, 4, 3, 
                      0, 4, 0, 0, 0, 0, 7, 7, 6, 7, 3, 0, 0, 3, 0, 0, 3, 2, 
                      3, 0, 6, 3, 3, 2}, 
                    1}}}, 
{"5,10,1", 
                x\ \((1 + x)\)\^2\ \((\(-3\) + x + x\^2)\)\ \((8 - 7\ x - 
                      6\ x\^2 + 2\ x\^3 + x\^4)\), {{{8, 4, 3, 4, 0, 0, 0, 0, 
                      0, 0, 7, 3, 7, 3, 3, 4, 3, 0, 0, 0, 3, 7, 2, 7, 0, 0, 
                      2, 3, 2}, 1}}}\)},{},
{  \({"5,12,768", \((\(-3\) + x)\)\^2\ \((1 + x)\)\^8\ \((3 + x)\), {{{6, 7, 6, 5, 7}, 
                    10}, "gS[5,{{1,6},{.4,6,1,-1.059}}]"}}, 
{"Icosahedron
 120", \((1 + x)\)\^5\ \((\(-5\) + x\^2)\)\^3, {"gS[5,{{1,3},{2,6,1,1.047},{1,3,1,3.15}},{{-1,8},{-2,4},{-3,6},{1,5,0},{2,6,0},{3,1,0}}]"}}, 
{"5,12,48", \((\(-2\) + 
                      x)\)\ \((\(-1\) + x)\)\^3\ x\^3\ \((2 + x)\)\^3\ \((4 + 
                      x)\), {"gS[5,{{1,12,1,-.785}},{{1,7,1,1,6},{1,4,2,2,6},{1,9,1,1,2},{3,7,1,1,2},{5,12,1,-1,2}}]"}}, 
{"5,12,24", \((\(-1\) + x)\)\^3\ \((1 + x)\)\^2\ \((3 + x)\)\^2\ \((\(-3\) + x\^2)\)\^2, {"gS[5,{{.545,4,2,2.37,-1},{1.,8,1,1.97}},{{4},12}]"}}, 
{"5,12,20", \((\(-1\) + x)\)\ x\ \((4 + 
                      x)\)\ \((1 - 4\ x - 4\ x\^2 + x\^3 + x\^4)\)\^2, {"gS[5,{{0.2,2,0},{.6,5,0,-.315},{1.,5.,-1,.3 3.1416}}]"}}, 
{"5,12,6", \((1 + x)\)\^4\ \((3 + 
                      x)\)\ \((3 - 5\ x - x\^2 + x\^3)\)\^2, {"gS[5,{{1,6},{.4,6,3,-1.059}},{{7,8,2,2,3}}]"}}, 
{"5,12,4", \((\(-1\) + 
                      x)\)\ \((1 + x)\)\^4\ \((\(-13\) - 9\ x + x\^2 + 
                      x\^3)\)\ \((\(-1\) - 5\ x + x\^2 + 
                      x\^3)\), {"gS[5,{{.545,4,2,2.37,-1},{1.,8,1,1.97}},{{2},12}]"}}\)},{},
{  \({"5,14,1792", \((1 + x)\)\^7\ \((1 - 9\ x - x\^2 + x\^3)\)\^2, {"gS[5,{{1,7,1,.225},{.4,7,1,-.67 }}]"}}, {"5,14,1", 
                x\ \((1 + x)\)\ \((2 + x)\)\ \((\(-12\) + 36\ x + 104\ x\^2 - 
                      244\ x\^3 - 152\ x\^4 + 190\ x\^5 + 97\ x\^6 - 
                      36\ x\^7 - 18\ x\^8 + 2\ x\^9 + 
                      x\^10)\), {"gS[5,{{1,9},{.4,5}},{{3,6,1,1,3},{1,7,1,2,3},{5,4,0}}]"}}, 
{"P14 512", \((\(-1\) + 
                      x)\)\ x\^4\ \((1 + x)\)\^2\ \((2 + x)\)\^3\ \((\(-2\) - 
                      13\ x - 2\ x\^2 + 
                      x\^3)\), {"gS[5,{{1,12,1,1.31},{.1,2}},{{-1,12,5,-5,2},{1,6,6,6,2},{1,3,1,1,4},{7,9,1,1,4},{1,5,1,1,2},{2,5,6,6,2},{3,5,6,6,2},{7,11,1,1,2}}]", 
"gS[5,{{1,7},{.2,7,1,0,-1}},{{1,2,7},{2,3,5},{8,2,7},{9,3,5},{1,7,0}}]"}}\)},{},
{  \({"5,16,4096", \((\(-1\) + x)\)\^2\ \((1 + x)\)\^8\ \((3 + x)\)\ \((\(-7\) - 2\ x + x\^2)\)\^2, {"gS[5,{{1,8,1,.3925},{.414,8,1,-.4}}]"}}, 
{"5,16,2304", \((\(-3\) + 
                      x)\)\ \((\(-1\) + x)\)\^6\ \((1 + x)\)\^6\ \((3 + 
                      x)\)\ \((5 + x)\), {{{3, 5, 7, 9, 11, 13}, 
                    7}}}, {"Clebsch 1920", \((\(-1\) + x)\)\^10\ \((3 + x)\)\^5, {"gS[5,{{3,5,1,-Pi/10.},{1,5,3,-Pi/10.},{2,5,0,-2.2},{0,1}},{{-6,11},{-7,12},{-8,13},{-9,14},{-10,15},{1,10,-8},{1,12,-11},{2,5,-4}}]"}}\)},{},{},{},
{  \({"5,20,20", \((1 + x)\)\ \((\(-5\) + 
                      x\^2)\)\ \((\(-5\) - 15\ x - 10\ x\^2 + x\^4)\)\^2\ \((5 - 5\ x - 4\ x\^2 + 2\ x\^3 + x\^4)\)\^2, {"gS[5,{{.3,5,1,-2.2},{1,10,1,-2.84},{.6,5,1,-.315}},{{-1,7,1,2,5},{2,7,1,2,5},{1,14,0}}]"}}\)},{},{  \({"5,22,5", \((20 + 24\ x - 12\ x\^2 - 12\ x\^3 + 
                      x\^4 + x\^5)\)\ \((1 - 16\ x - 22\ x\^2 + 56\ x\^3 + 28\ x\^4 - 28\ x\^5 - 13\ x\^6 + 2\ x\^7 + x\^8)\)\^2, {"gS[5,{{.1,1},{1.1,5,1,-1},{2,10},{.7,5,1,3.5},{0,1}},{{1,8,0,2,5}}]"}}\)},{},
{  \({"SnubCube 512", \((1 + x)\)\^4\ \((\(-6\) - 2\ x + x\^2)\)\^3\ \((\(-2\) + 2\ x + x\^2)\)\^2\ \((\(-2\) - 4\ x + x\^2 + x\^3)\)\^3, {"gS[5,{{1,4,1,1.2},{1.5,8,1,-5},{2,8,1,-2.3},{3.5,4,1}}]"}}\)},{},{},{},{},{},{  \({"Wong 120", \((\(-1\) + x)\)\^5\ \((1 + x)\)\^2\ \((\(-5\) + x\^2)\)\^3\ \((\(-5\) + x + x\^2)\)\^8, {"gS[5,{{1,30}},{{-14,-8,15,8,14,15,4,12,-4,-12,4,8,-4,-12,4,12,-4,-8},5}]"}}, {"Meringer 96", \((\(-2\) + x)\)\^9\ x\ \((2 + x)\)\^3\ \((3 + x)\)\^2\ \((\(-4\) + x + x\^2)\)\^3\ \((\(-2\) + 2\ x + x\^2)\)\^4, {"gS[5,{{1,30}},{{15,-8,12,-7,-4,15,-11,12,-5,9,12,5,15,5,14,11,7,8,-9,-8},3}]"}}, {"Foster 5,5
-cage 30", \((\(-2\) + x)\)\^4\ \((1 + 
                      x)\)\ \((\(-5\) + x\^2)\)\^2\ \((\(-4\) + 2\ x + x\^2)\)\^2\ \((11 - 7\ x - 6\ x\^2 + 2\ x\^3 + x\^4)\)\^4, {{{12, 9, 12, 6, 
                      12, 9, 12, 6, 12, 9, 12, 6}, 
                    5}}}, {"Robertson
-Wegner 20", \((\(-2\) + x)\)\^8\ \((1 + x)\)\ \((3 + x)\)\^4\ \((11 - 7\ x - 6\ x\^2 + 2\ x\^3 + x\^4)\)\^2\ \((5 - 5\ x - 4\ x\^2 + 2\ x\^3 + x\^4)\)\^2, {"gS[5,{{1,30}},{{12,6,9,6,9,6,15,-9,12,15,12,-9},5}]"}}\)},{},
{  \({"Wells?1920", \((\(-1\) + x)\)\^10\ \((3 + x)\)\^5\ \((\(-5\) + x\^2)\)\^8, {"gS[5,{{1.,25.,-1},{.85,5,1,-.315},{0.2,-2,0,-.3,-1}},{{-2,26,5,1,5},{-3,26,5,1,5},{31,3,0,5,5},{32,1,0,5,5},{26,9,1,5,4},{30,4},{30,18,-1,-5,5},{26,23},{2,7,5,5,4},{2,22},{2,10,5,5,4},{5,22},{5,13,5,5,3},{20,3,5,5,2},{19,1},{4,11,5,5,3},{6,24},{5,9,5,5,4},{25,4}}]"}}\)},{},{},{},{  {"Sylvester",(((-2 + x))^16) (((1 + x))^10) (((3 + x))^9),{"subGraph[{1,2},-1.,{},graphNo[7,50,1]]"}}}};

6 regular

x =. ;graph6Data={{},{},{},{},{},{},
{  \({"7,7,5040", \ \(-\((1 + x)\)\^6\), {{{2, 3}, 
                    8}, 
"gS[6,{{1,6,1,.523},{.4,1}},{{2,3},6}]", 
"gS[6,{{.2,2,1},{1,5,1,-.314}},{{3,2,5},{1,1,0}}]"}}\)},
{  \({"16-Cell,384", \ 
                x\^4\ \((2 + x)\)\^3, {{{2, 3, 5}, 
                    8}, 
"gS[6,{{.3,2},{1,6}},{{3,2,5},{1,1,0}}]", "gS[6,{{.2,3,1,-.515},{1,5,1,-.315}},{{5,2,5},{6,2,5}}]", "gS[6,{{1,2,1,-.75},{2,4,1,0},{1,2,1,.75}},{{1,7,1,1,2},{1,8,1,-1,2},{3,2,0}}]", "gS[6,{{.3,2,1,-1.57},{1,6}},{{0,0,3,4,2,-2,4,-3},1}]"}}\)},
{  \({"6,9,1296", \(-x\^6\)\ \((3 + x)\)\^2, {{{2, 4}, 
                    9}}}, {"6,9,18", \(-x\^2\)\ \((\(-3\) + 3\ x\^2 + x\^3)\)\^2, 
{{{1, 2, 5}, {1, 3, 
                      8}}, "gS[6,{{1,8},{.2,1}},{{3,8,1,0,3},{2,6,0,1,2},{1,3,1,1,7},{1,4,1,1,7}}]", "gS[6,{{1,7,1,.675},{.2,2}},{{3,9,1,-1,2},{1,3,1,1,7},{1,5,1,1,7}}]", "gS[6,{{.35,3,1,.525,1},{1,6}},{{4,2,5}}]", 
{{1, 3, 
                      10}, {1, 4, 6}, {2, 4, 2}, {3, 4, 2}}}}\)},
{  \({"6,10,120", \((\(-1\) + x)\)\^4\ \((2 + x)\)\^5, {"gS[6,{{1,5,3,1.9,-1},{1.,5}}]"}}, {"6,10,32", \ \((\(-1\) + 
                      x)\)\ x\^2\ \((1 + x)\)\^2\ \((2 + x)\)\ \((\(-4\) - 
                      4\ x + 3\ x\^2 + x\^3)\), {{{2, 3, 5}, 
                    10}}}, 
{"6,10,20", \((2 + 
                      x)\)\ \((\(-1\) - x + x\^2)\)\^2\ \((1 + 3\ x + x\^2)\)\^2, {{{2, 3, 7}, 
                    10}}}, 
{"6,10,6", \((\(-1\) + x)\)\^2\ \((2 + x)\)\ \((\(-1\) + 3\ x\^2 + x\^3)\)\^2, {{{2, 3, 4}, 10}}}\)},
{  \({"6,11,22,1", \(-\((1 + x - 4\ x\^2 - 3\ x\^3 + 3\ x\^4 + x\^5)\)\^2\), {{{1, 2, 4}, 11}, {{3, 6}, 
                    11}}}, 
{"6,11,22,2", \(-\((1 - 10\ x - 15\ x\^2 - 3\ x\^3 + 3\ x\^4 + x\^5)\)\^2\), {{{3, 4}, 11}, {{3, 2}, 
                    11}}}, 
{"6,11,4", \(-\((1 + x)\)\^2\)\ \((\(-3\) + x\^2)\)\^2\ \((\(-2\) + 2\ x + x\^2)\)\^2, {"gS[6,{{1,6},{.45,5,3,-.8}},{{1,3,1,3,2}}]"}}, 
{"6,11,2,1", \(-\((\(-1\) + x)\)\)\ \((2 + 
                      x)\)\ \((\(-5\) - 3\ x + 2\ x\^2 + x\^3)\)\ \((2 + x - 
                      8\ x\^2 - 3\ x\^3 + 3\ x\^4 + 
                      x\^5)\), {"gS[6,{{1.,11}},{{2,3,4,5,2,5,-5,4,4,3,3,9},5}]"}}, 
{"6,11,2,2", \(-x\)\ \((2 + x)\)\ \((\(-2\) + 
                      x\^2)\)\ \((1 + 3\ x + x\^2)\)\ \((2 - 7\ x\^2 + x\^3 + 
                      x\^4)\), {"gS[6,{{1.,9,1,.87},{.3,2}},{{2,4},10}]"}}\)},
{  \({"6,12,48", \((\(-2\) + x)\)\^3\ x\^2\ \((2 + x)\)\^6, {"gS[6,{{1,3,1,-.525},{2,6,1,-1.05},{1.5,3,1,.525}}]", "gS[6,{{1,6},{.4,6,2,-1.55}}]"}}, {"6,12,36", \((\(-1\) + x)\)\^2\ \((2 + x)\)\^5\ \((\(-3\) - x + x\^2)\)\^2, {"gS[6,{{1,6},{.2,6,3,-1.55}},{{7,8,2,2,3}}]"}}, {"6,12,24", \((\(-2\) + 
                      x)\)\ x\^2\ \((1 + x)\)\^2\ \((3 + x)\)\^2\ \((\(-3\) + x\^2)\)\^2, {{{2, 4, 0, 2, 4}, 9}}}, 
{"6,12,16", 
                x\^6\ \((\(-4\) + 2\ x + x\^2)\)\ \((\(-8\) - 4\ x + 
                      4\ x\^2 + x\^3)\), {{{3, 4, \(-4\), \(-3\), 6}, 
                    9}}}, 
{"6,12,12", \((\(-2\) + 
                      x)\)\ \((1 + x)\)\^2\ \((2 + x)\)\^2\ \((\(-3\) - 5\ x + x\^2 + x\^3)\)\^2, {"gS[6,{{1.,9,1,-.175},{.32,3,0,-1.2}},{{1,3,1,1,7},{8,1,1,1,2},{-3,10,3,1,3},{10,11},{11,12},{10,12},{3,6},{6,9},{3,9}}]"}}, {"6,12,8,1", 
                x\^2\ \((2 + x)\)\^5\ \((\(-2\) - 2\ x + x\^2)\)\^2, {"gS[6,{{1,8,1,.39},{.4,4,1,-.79}},{{1,3,1,1,8},{1,7,1,1,2}}]", "gS[6,{{1,6},{.45,6,1,-1.05}}]"}}, 
{"6,12,8,2", \((\(-2\) + x)\)\ \((\(-1\) + x)\)\ \((1 + x)\)\^4\ \((3 + 
                      x)\)\ \((\(-4\) + x + x\^2)\)\^2, {"gS[6,{{.8,4,0,3.142,-1},{1.,8,1,1.97}},{{4},12}]"}}\)},
{  \({"6,13,78", \(-\((\(-3\) + x + x\^2)\)\^6\), {{{3, 
                      4}, 13}}}, 
{"6,13,26", \(-\((\(-1\) - 12\ x - 34\ x\^2 - 30\ x\^3 - 6\ x\^4 + 3\ x\^5 + x\^6)\)\^2\), {{{4, 5}, 13}}}\)},
{  \({"6,14,10080", \((\(-1\) + x)\)\^6\ \((1 + x)\)\^6\ \((6 + x)\), {{{3, 5, 5, 3}, 
                    7}}}, {"6,14,28,1", \((1 + x)\)\^6\ \((2 + 
                      x)\)\ \((1 - 9\ x - x\^2 + x\^3)\)\^2, {"gS[6,{{1,7},{.2,7,1,-1.4}}]"}}, 
{"6,14,28,2", \((2 + x)\)\ \((1 - 2\ x - x\^2 + x\^3)\)\^2\ \((\(-13\) - 4\ x + 3\ x\^2 + x\^3)\)\^2, {{{3, 4, 4, 3}, 
                    7}}}, 
{"6,14,28,3", \((2 + 
                      x)\)\ \((1 - 2\ x - x\^2 + x\^3)\)\^2\ \((1 - 4\ x + 3\ x\^2 + x\^3)\)\^2, {{{2, 5, 5, 2}, 
                    7}}}, 
{"6,14,2", \((\(-2\) + 
                      x)\)\ x\ \((2 + x)\)\ \((\(-1\) + x + 
                      x\^2)\)\ \((\(-1\) + 3\ x + x\^2)\)\ \((\(-8\) - 7\ x + 
                      x\^2 + x\^3)\)\ \((2 - 5\ x + x\^2 + 
                      x\^3)\), {"gS[6,{{1,10},{.2,4,1,-.77}},{{2,10,3,-3,4},{10,6,-4,-4,2},{2,4,6,6,2},{3,6,0}}]"}}\)},
{  \({"6,15,77760" (*Highest\ in\ this\ collection*) , \(-x\^10\)\ \((\(-9\) + 3\ x + x\^2)\)\^2, {{{4, \(-6\), 6, \(-4\)}, 
                    15}, \ "gS[2,{{.63,5,1,-Pi/10.},{2,5,1,-Pi/10.},{3,5,1,-Pi/10.}},{{1,-7,1,1,4},{1,-10},{2,-6,1,1,4},{5,-6},{6,-12,1,1,4},{10,-11},{7,-11,1,1,4},{6,-15},{1,-12,1,1,4},{5,-11},{2,-11,1,1,4},{1,-15}}]"}}, {"6,15,320", \(-x\^6\)\ \((\(-4\) - 2\ x + x\^2)\)\^2\ \((5 + 5\ x + x\^2)\)\^2, {"gS[4,{{1.15,5,1,Pi/10.},{3,5,1,3Pi/10.},{2,5,1,Pi/10.}},{{1,-6,1,1,5},{1,-7,1,1,4},{5,-6},{1,-12,1,1,4},{5,-11},{2,-11,1,1,4},{1,-15}}]"}}, {"6,15,30,1", \(-x\^2\)\ \((1 + 3\ x + x\^2)\)\^2\ \((\(-5\) - 15\ x - 10\ x\^2 + x\^4)\)\^2, {{{6, 7, \(-7\), \(-6\)}, 
                    20}}}, 
{"6,15,30,2", \(-\((\(-1\) + x)\)\^4\)\ x\^2\ \((\(-5\) - 10\ x + 5\ x\^3 + x\^4)\)\^2, {{{3, 5}, 
                    30}}}\)},
{  \({"6,16,12288", \((\(-2\) + x)\)\^3\ x\^8\ \((2 + x)\)\^3\ \((6 + x)\), {{{7, 5, 3, \(-3\), \(-5\), 7}, 
                    16}}}, {"6,16,96", \((\(-2\) + 
                      x)\)\ \((\(-1\) + x)\)\^2\ x\^4\ \((1 + x)\)\^2\ \((2 + 
                      x)\)\ \((6 + x)\)\ \((\(-3\) + x\^2)\)\^2, {{{3, 5, 7}, 
                    16}}}, {"6,16,64,1", \((2 + x)\)\^3\ \((\(-8\) + x\^2)\)\^2\ \((\(-2\) + x\^2)\)\^4, {{{7, 10, 7, 10, 7}, 
                    10}}}, {"6,16,64,2", \((\(-2\) + x)\)\^2\ x\^4\ \((2 + x)\)\^5\ \((\(-8\) + x\^2)\)\^2, {{{5, 10, 7, 9, 6, 11}, 
                    8}}}, {"6,16,32", 
                x\^4\ \((2 + x)\)\^3\ \((\(-4\) - 16\ x - 12\ x\^2 + x\^4)\)\^2, {{{5, 10, 5, 10, 5}, 10}}}\)},{},{},{  {"6,19,288", -(x^6) (6 - 47 x + 55 (x^2) + 
                      10 (x^3) - 21 (x^4) + (x^5) + (x^6)) (-6 + 
                      21 x + 15 (x^2) - 26 (x^3) - 5 (x^4) + 5 (x^5) + 
                      (x^6)), {{{8, 6, 4}, 20}}}},
{},{},{},{},{},{},{},{},{},{},{},{},{},{},{},{},{},{},{},{},{},{},{  {"6,42",(((-2 + x))^21) (((1 + x))^6) (((3 + x))^14),{"subGraph[{1},-1.,{},graphNo[7,50,1]]"}}}};

7 regular

x =. ;graph7Data={{},{},{},{},{},{},{},
{  {"7,8,40320", (((1 + x))^7), {{{2, 3, 4}, 
                    8}, "gS[7,{{1,7},{0,1}},{{2,3,4},8}]"}}},{},
{  \({"7,10,200", \((3 + 
                      x)\)\ \((\(-1\) + x + x\^2)\)\^4, {{{3, \(-2\), 5, 
                      3, \(-2\), 5}, 
                    5}, "gS[7,{{1,5},{.4,5,1}}]"}}, {"7,10,96", \((\(-1\) + x)\)\^2\ x\^2\ \((1 + x)\)\^2\ \((2 + x)\)\^2\ \((3 + 
                      x)\), {{{2, \(-2\), 5, 4, \(-4\), 3, \(-2\), 5}, 
                    5}}}, {"7,10,20", \((\(-1\) + 
                      x)\)\ \((\(-1\) + x + x\^2)\)\^2\ \((1 + 3\ x + x\^2)\)\^2, {{{2, 3, 4, \(-4\), \(-3\), \(-2\)}, 
                    6}, {{8, 2, 2, 0, 0, 0, 0, 0, 0, 0, 4, 3, 3, 2, 0, 0, 0, 
                      0, 0, 0, 5, 4, 4, 4, 4, 0, 2, 0, 0, 0, 6, 5, 5, 5, 5, 
                      4, 3, 2, 0, 0, 7, 6, 6, 6, 6, 5, 4, 3, 2, 0, 8, 2, 2}, 
                    1}}}\)},{},
{  \({"7,12,4608", \((\(-1\) + x)\)\^4\ \((1 + x)\)\^6\ \((5 + x)\), {{{3, \(-5\), 6, \(-5\), \(-3\)}, 
                    10}}}, {"7,12,48,1", \((\(-2\) + x)\)\^2\ \((\(-1\) + x)\)\^2\ \((1 + x)\)\ \((2 + x)\)\^6, {{{4, 5, 6, \(-5\), \(-4\)}, 
                    12}}}, {"7,12,48,2", \((\(-2\) + x)\)\^2\ x\^4\ \((1 + x)\)\ \((2 + x)\)\^2\ \((3 + x)\)\^2, {{{2, \(-5\), 6, \(-5\), \(-2\)}, 
                    10}, "gS[7,{{1,6},{.4,6,1,.1}}]"}}, {"7,12,24,1", \((\(-3\) + 
                      x)\)\ x\^2\ \((1 + x)\)\^2\ \((2 + x)\)\^2\ \((\(-2\) + 2\ x + x\^2)\)\^2, {{{2, \(-4\), 6, \(-4\), \(-2\)}, 
                    10}}}, 
{"7,12,24,2", \((\(-1\) + x)\)\^4\ \((1 + x)\)\^3\ \((1 + 4\ x + x\^2)\)\^2, {{{3, \(-4\), 
                      6, \(-4\), \(-3\)}, 
                    10}}}, {"7,12,24,3", \((\(-1\) + x)\)\^2\ \((1 + x)\)\^3\ \((3 + x)\)\^2\ \((\(-3\) + x\^2)\)\^2, {{{2, \(-3\), 
                      6, \(-3\), \(-2\)}, 
                    10}}}, {"7,12,16", \((\(-2\) + x)\)\ x\^3\ \((1 + 
                      x)\)\ \((2 + x)\)\^3\ \((3 + x)\)\ \((\(-4\) - x + 
                      x\^2)\), {{{3, \(-4\), 5, 4, \(-3\)}, 
                    12}}}, {"7,12,2,1", \((1 + 3\ x - 7\ x\^2 - 2\ x\^3 + 
                      3\ x\^4 + x\^5)\)\ \((1 + 16\ x - 4\ x\^2 - 21\ x\^3 - 
                      3\ x\^4 + 4\ x\^5 + x\^6)\), {{{3, 2, \(-4\), 4, 5, 3, 
                      5}, 11}}}, {"7,12,2,2", 
                x\ \((2 + x)\)\ \((5 - 5\ x - 5\ x\^2 + 2\ x\^3 + 
                      x\^4)\)\ \((4 + 6\ x - 11\ x\^2 - 4\ x\^3 + 3\ x\^4 + 
                      x\^5)\), {{{2, 3, 4, \(-5\), 6}, 
                    12}, "gS[7,{{1,9},{.7,3,1,-1.2}},{{5,3,6,2,4,3,5},11}]"}}, {"7,12,2,3", \((\(-2\) + x)\)\ \((\(-1\) + x)\)\^2\ \((1 + 
                      x)\)\ \((2 + x)\)\^4\ \((\(-4\) - 4\ x + 2\ x\^2 + 
                      x\^3)\), {"gS[7,{{1,7},{.8,5,1,-1.2}},{{4,3,-3,3,4},6}]"}}, {"7,12,1,1", 
                5 + 7\ x - 79\ x\^2 - 131\ x\^3 + 121\ x\^4 + 267\ x\^5 + 
                  36\ x\^6 - 114\ x\^7 - 49\ x\^8 + 7\ x\^9 + 7\ x\^10 + 
                  x\^11, {{{3, \(-4\), 2, \(-4\), \(-3\)}, 
                    10}}}, {"7,12,1,2", \((2 + x)\)\ \((\(-1\) - 6\ x - 
                      2\ x\^2 + 3\ x\^3 + x\^4)\)\ \((\(-2\) + 5\ x + 
                      9\ x\^2 - 10\ x\^3 - 7\ x\^4 + 2\ x\^5 + x\^6)\), {{{3, 
                      2, 4, 5, 6}, 
                    12}}}, {"7,12,1,3", \((\(-1\) + x)\)\^2\ \((2 + 
                      x)\)\ \((\(-2\) + 2\ x + x\^2)\)\ \((4 + 8\ x - 
                      16\ x\^2 - 18\ x\^3 + 2\ x\^4 + 5\ x\^5 + 
                      x\^6)\), {{{2, 3, 4, \(-5\), \(-4\)}, 12}}}\)},{},
{  \({"7,14,?", 
                x\^12\ \((7 + x)\), {{{3, 5, 7}, 
                    14}}}, {"7,14,28,1", \((3 + 
                      x)\)\ \((\(-7\) - 7\ x + x\^3)\)\^2\ \((\(-1\) - x + 2\ x\^2 + x\^3)\)\^2, {{{5, 6, 7}, 14}, {{3, 4, 7}, 
                    14, {{5, 6, 7, \(-6\), \(-5\)}, 
                      12}}, "gS[7,{{1,7},{.7,7,1,-1.795}}]"}}, 
{"7,14,28,2", \((\(-1\) + 
                      x)\)\ x\^6\ \((\(-8\) - 4\ x + 4\ x\^2 + x\^3)\)\^2, {{{2, 4, 7}, 14}, {{3, 4, 6, \(-6\), \(-4\), \(-3\)}, 7}}}, {"7,14,28,3", 
                x\^6\ \((3 + 
                      x)\)\ \((\(-8\) - 8\ x + 2\ x\^2 + x\^3)\)\^2, {{{2, 3, 
                      7}, 14}, {{4, 5, 7}, 
                    14}}}, {"7,14,28,4", \((3 + 
                      x)\)\ \((7 - 7\ x + x\^3)\)\^2\ \((\(-1\) - x + 2\ x\^2 + x\^3)\)\^2, {{{2, 5, 7}, 14}, {{3, 6, 7}, 
                    14}}}, {"7,14,28,5", \((\(-1\) + 
                      x)\)\ \((\(-7\) - 7\ x + x\^3)\)\^2\ \((\(-1\) + 3\ x + 4\ x\^2 + x\^3)\)\^2, {{{2, 6, 7}, 
                    14}}}, {"7,14,28,6", \((\(-1\) + 
                      x)\)\ \((7 - 7\ x + x\^3)\)\^2\ \((\(-1\) + 3\ x + 4\ x\^2 + x\^3)\)\^2, {{{4, 6, 7}, 
                    14}}}, {"7,14,14,1", \((3 + 
                      x)\)\ \((\(-7\) + 21\ x + 14\ x\^2 - 15\ x\^3 - 8\ x\^4 + 2\ x\^5 + x\^6)\)\^2, {"gS[7,{{1,7},{.5,7,2,-1.795}}]"}}, {"7,14,14,2", \((3 + 
                      x)\)\ \((7 + 28\ x + 14\ x\^2 - 15\ x\^3 - 8\ x\^4 + 2\ x\^5 + x\^6)\)\^2, {"gS[7,{{1,7},{.5,7,3,-1.795}}]"}}, {"7,14,14,3", \((\(-1\) + 
                      x)\)\ \((7 + 14\ x - 7\ x\^2 - 22\ x\^3 - 4\ x\^4 + 4\ x\^5 + x\^6)\)\^2, {{{5, 4, \(-6\), 6, \(-4\), \(-5\)}, 
                    7}}}, {"7,14,4", \((1 + 
                      x)\)\ \((1 + 3\ x + x\^2)\)\^2\ \((3 + x - 6\ x\^2 - 
                      x\^3 + x\^4)\)\ \((\(-7\) - 19\ x - 10\ x\^2 + x\^3 + 
                      x\^4)\), {{{2, 3, 4, \(-3\), \(-2\)}, 
                    14}}}, 
{"7,14,2,1", \((1 + 
                      x)\)\ \((7 + 9\ x - 7\ x\^2 - 7\ x\^3 + x\^4 + 
                      x\^5)\)\ \((5 + 23\ x + 2\ x\^2 - 60\ x\^3 - 48\ x\^4 - 
                      4\ x\^5 + 5\ x\^6 + x\^7)\), {{{3, 4, \(-5\), 5, 3}, 
                    14}}}, 
{"7,14,2,2", \((1 + 
                      x)\)\ \((3 + x)\)\ \((7 + 9\ x - 7\ x\^2 - 7\ x\^3 + 
                      x\^4 + x\^5)\)\ \((7 + 32\ x + 10\ x\^2 - 22\ x\^3 - 
                      10\ x\^4 + 2\ x\^5 + x\^6)\), {{{3, 4, 
                      5, \(-4\), \(-3\)}, 
                    14}}}, {"7,14,2,3", \((3 + 
                      x)\)\ \((7 + 28\ x + 14\ x\^2 - 15\ x\^3 - 8\ x\^4 + 2\ x\^5 + x\^6)\)\^2, {{{2, 4, 5, \(-4\), \(-2\)}, 14}}}, {"7,14,2,3", 
                x\^2\ \((3 + x)\)\ \((12 + 11\ x - 14\ x\^2 - 10\ x\^3 + 
                      2\ x\^4 + x\^5)\)\ \((4 + 3\ x - 10\ x\^2 - 6\ x\^3 + 
                      2\ x\^4 + x\^5)\), {{{2, 3, 5, \(-3\), \(-2\)}, 
                    14}}}, 
{"7,14,1,1", 
                7 - 61\ x - 153\ x\^2 + 341\ x\^3 + 497\ x\^4 - 543\ x\^5 - 
                  602\ x\^6 + 336\ x\^7 + 328\ x\^8 - 74\ x\^9 - 80\ x\^10 + 
                  7\ x\^12 + 
                  x\^13, {{{3, 4, 6, 7, \(-5\)}, 
                    12}}}, 
{"7,14,1,2", 
                28 - 172\ x - 245\ x\^2 + 653\ x\^3 + 682\ x\^4 - 798\ x\^5 - 
                  737\ x\^6 + 407\ x\^7 + 362\ x\^8 - 78\ x\^9 - 82\ x\^10 + 
                  7\ x\^12 + 
                  x\^13, {{{3, \(-4\), 7, 5, 6}, 
                    14}}}, 
{"7,14,1,3", 
                112 - 608\ x - 680\ x\^2 + 1564\ x\^3 + 1231\ x\^4 - 
                  1463\ x\^5 - 1016\ x\^6 + 604\ x\^7 + 426\ x\^8 - 
                  98\ x\^9 - 88\ x\^10 + 7\ x\^12 + 
                  x\^13, {{{3, 5, 4, \(-6\), 7}, 
                    14}}}, 
{"7,14,1,4", 
                x\ \((\(-40\) - 36\ x + 312\ x\^2 + 180\ x\^3 - 624\ x\^4 - 
                      364\ x\^5 + 424\ x\^6 + 280\ x\^7 - 92\ x\^8 - 
                      80\ x\^9 + 7\ x\^11 + x\^12)\), {{{3, 5, 6, \(-6\), 7}, 
                    13}}}\)},{},
{  \({"7,16,4096", \((\(-3\) + x)\)\^2\ \((1 + x)\)\^9\ \((\(-7\) + 2\ x + x\^2)\)\^2, {{{4, 7, 8}, 
                    16}}}, 
{"7,16,64,1", \((1 + x)\)\^3\ \((\(-7\) - 2\ x + x\^2)\)\^2\ \((\(-1\) + 2\ x + x\^2)\)\^4, {{{6, 
                      7, 8}, 16}, {{2, 7, 8}, 
                    16}}}, {"7,16,64,2", \((\(-1\) + x)\)\^2\ \((1 + x)\)\^7\ \((3 + x)\)\^2\ \((\(-7\) - 2\ x + x\^2)\)\^2, {"gS[7,{{1.,8},{.6,8,1,-1.57}}]"}}, 
{"7,16,32,1", \((\(-1\) + x)\)\^4\ \((1 + x)\)\^3\ \((\(-31\) - 36\ x - 6\ x\^2 + 4\ x\^3 + x\^4)\)\^2, {{{5, 6, 8}, 16}, {{2, 3, 8}, 
                    16}}}, 
{"7,16,32,2", \((\(-3\) + 
                      x)\)\ \((\(-1\) + x)\)\^2\ \((\(-1\) + 2\ x + x\^2)\)\^2\ \((\(-1\) - 4\ x - 2\ x\^2 + 4\ x\^3 + x\^4)\)\^2, {{{4, 6, 8}, 
                    16}}}, 
{"7,16,32,3", \((\(-1\) + x)\)\^2\ \((5 + 
                      x)\)\ \((\(-1\) - 2\ x + x\^2)\)\^2\ \((\(-1\) - 4\ x + 2\ x\^2 + 4\ x\^3 + x\^4)\)\^2, {{{5, 7, 8}, 
                    16}}}, 
{"7,16,32,4", \((\(-1\) + x)\)\^4\ \((1 + x)\)\^3\ \((1 - 4\ x - 6\ x\^2 + 4\ x\^3 + x\^4)\)\^2, {{{2, 
                      5, 8}, 16}, {{3, 6, 8}, 
                    16}}}, 
{"7,16,32,5", \((\(-3\) + x)\)\^2\ \((1 + x)\)\^5\ \((1 - 12\ x - 2\ x\^2 + 4\ x\^3 + x\^4)\)\^2, {{{4, 
                      5, 8}, 16}}}, 
{"7,16,32,6", \((\(-3\) + 
                      x)\)\ \((3 + x)\)\^2\ \((\(-1\) - 2\ x + x\^2)\)\^2\ \((\(-1\) - 4\ x + 2\ x\^2 + 4\ x\^3 + x\^4)\)\^2, {{{2, 6, 8}, 
                    16}}}, 
{"7,16,32,7,", \((\(-3\) + 
                      x)\)\ \((\(-1\) + x)\)\^2\ \((\(-1\) + 2\ x + x\^2)\)\^2\ \((\(-17\) - 20\ x - 2\ x\^2 + 4\ x\^3 + x\^4)\)\^2, {{{2, 4, 8}, 
                    16}}}, {"7,16,16,1", \((3 + x)\)\ \((\(-5\) + 
                      x\^2)\)\ \((\(-1\) + 2\ x + x\^2)\)\^2\ \((1 - 8\ x - 8\ x\^2 + x\^4)\)\^2, {"gS[7,{{1.,8},{.6,8,2,-1.57}}]"}}, 
{"7,16,2,1", 
                x\ \((1 + x)\)\^5\ \((2 + x)\)\ \((3 + x)\)\ \((12 - 6\ x - 
                      3\ x\^2 + x\^3)\)\ \((28 - 12\ x\^2 + x\^4)\), {{{5, 6, 
                      7, \(-6\), \(-5\)}, 
                    16}}}, 
{"7,16,2,2", \((5 - 16\ x - 
                      7\ x\^2 + 30\ x\^3 - 2\ x\^4 - 11\ x\^5 + x\^6 + 
                      x\^7)\)\ \((1 - 15\ x - 3\ x\^2 + 67\ x\^3 - 8\ x\^4 - 
                      45\ x\^5 - 2\ x\^6 + 6\ x\^7 + x\^8)\), {{{5, 6, 
                      8, \(-6\), \(-3\)}, 
                    16}}}, 
{"7,16,1,1", 
                x\ \((1 + x)\)\^5\ \((2 + x)\)\ \((3 + x)\)\ \((12 - 6\ x - 
                      3\ x\^2 + x\^3)\)\ \((28 - 12\ x\^2 + x\^4)\), {{{2, 3, 
                      4, 6, 7}, 16}}}, 
{"7,16,1,2", 
                396 + 536\ x - 2876\ x\^2 - 3900\ x\^3 + 5688\ x\^4 + 
                  9228\ x\^5 - 2143\ x\^6 - 7451\ x\^7 - 1302\ x\^8 + 
                  2292\ x\^9 + 901\ x\^10 - 211\ x\^11 - 151\ x\^12 - 
                  7\ x\^13 + 7\ x\^14 + 
                  x\^15, {{{2, \(-3\), 4, \(-4\), 5}, 16}}}\)},{},
{  \({"7,18", \((\(-2\) + x)\)\^2\ \((\(-1\) + x)\)\^6\ \((1 + x)\)\^6\ \((2 + x)\)\^2\ \((7 + x)\), {{{5, 7, 9, \(-7\), \(-5\)}, 
                    17}}}, {"7,18,36,1", \((\(-4\) + x)\)\^2\ x\^2\ \((3 + x)\)\ \((\(-3\) + 3\ x\^2 + x\^3)\)\^2\ \((\(-1\) + 3\ x\^2 + x\^3)\)\^2, {{{3, 6, 9, \(-6\), \(-3\)}, 
                    11}}}, {"7,18,36,2", \((\(-1\) + x)\)\^2\ \((3 + x)\)\^3\ \((9 - 9\ x + x\^3)\)\^2\ \((\(-1\) - 3\ x + x\^3)\)\^2, {{{3, 4, 
                      9, \(-4\), \(-3\)}, 
                    14}}}, {"7,18,36,3", \((\(-1\) + x)\)\^7\ \((2 + x)\)\^4\ \((\(-19\) - 9\ x + 3\ x\^2 + x\^3)\)\^2, {{{2, 4, 9, \(-4\), \(-2\)}, 
                    14}}}\)},{},
{  },{},
{  },{},
{  \({"Klein,336", \((1 + x)\)\^7\ \((\(-7\) + x\^2)\)\^8, {{{\(-2\), \(-6\), \(-8\), 
                      12, \(-6\), \(-7\), \(-7\), \(-11\), \(-4\), \(-7\), 
                      11, 11, \(-10\), \(-8\), \(-11\), 12, \(-5\), \(-9\), 
                      10, 10, \(-11\), \(-10\), 7, 10, \(-9\), \(-10\), 8, 6, 
                      11, \(-11\), 6, 9, 9, 11, 7, 5, 4, 8, 2, 7}, 
                    3}}}, 
{"7,24", \((\(-5\) + 
                      x)\)\ \((\(-1\) + x)\)\^10\ \((1 + x)\)\^10\ \((5 + 
                      x)\)\ \((7 + x)\), {{{3, 5, 7, 9, 
                      11, \(-11\), \(-9\), \(-7\), \(-5\), \(-3\)}, 11}}}\)},{},
{  },{},{  },{},
{  \({"7,30", 
                x\^2\ \((2 + x)\)\^2\ \((3 + 
                      x)\)\ \((\(-1\) + x + x\^2)\)\^4\ \((31 + 37\ x - x\^2 - 7\ x\^3 + x\^4)\)\^2\ \((\(-5\) - 5\ x + 5\ x\^2 + 5\ x\^3 + x\^4)\)\^2, {"gS[7,{{2,15,1},{1,15,1,-.8}}]"}}\)},{},
{  \({"7,32a", \((\(-5\) + 
                      x)\)\ \((\(-1\) + x)\)\^2\ \((1 + x)\)\^2\ \((5 + 
                      x)\)\ \((7 + 
                      x)\)\ \((\(-1\) - 2\ x + x\^2)\)\^2\ \((\(-1\) + 2\ x + x\^2)\)\^2\ \((\(-1\) + 4\ x + 2\ x\^2 - 4\ x\^3 + x\^4)\)\^2\ \((\(-1\) - 4\ x + 2\ x\^2 + 4\ x\^3 + x\^4)\)\^2, {{{3, 5, 7, 9, 
                      11, \(-11\), \(-9\), \(-7\), \(-5\), \(-3\)}, 
                    15}}}, {"7,32b", \((\(-5\) + 
                      x)\)\ \((\(-1\) + x)\)\^2\ \((1 + x)\)\^2\ \((5 + 
                      x)\)\ \((7 + 
                      x)\)\ \((\(-1\) - 2\ x + x\^2)\)\^2\ \((\(-1\) + 2\ x + x\^2)\)\^2\ \((7 - 6\ x\^2 + x\^4)\)\^4, {{{5, 7, 9, 11, 
                      13, \(-13\), \(-11\), \(-9\), \(-7\), \(-5\)}, 15}}}\)},{},{  },{},
{  \({"7,36", \((\(-5\) + 
                      x)\)\ \((\(-2\) + x)\)\^4\ \((\(-1\) + x)\)\^6\ \((1 + x)\)\^6\ \((2 + x)\)\^4\ \((5 + x)\)\ \((7 + 
                      x)\)\ \((\(-73\) + 63\ x\^2 - 15\ x\^4 + x\^6)\)\^2, {{{5, 7, 9, 11, 13, \(-13\), \(-11\), \(-9\), \(-7\), \(-5\)}, 18}}}\)},{},{  },{},{  },{},{  },{},{  },{},{  },{},{  },{},{  \({"Hoffman
Singleton", \((\(-2\) + x)\)\^28\ \((3 + x)\)\^21, {{{4, 5, 7, 4, 7, 4, 6, 8, 5, 18, 8, 13, 14, 15, 24, 12, 17, 22, 
                      16, 28, 19, 24, 18, 23, 34, 16, 27, 32, 26, 31, 22, 34, 
                      28, 33, 37, 26, 36, 38, 0, 0}, 
                    5}}}, {"7,50,?", \((3 + 
                      x)\)\ \((\(-1\) + x + x\^2)\)\^4\ \((1999 + 7515\ x + 9480\ x\^2 + 3135\ x\^3 - 2430\ x\^4 - 1699\ x\^5 + 105\ x\^6 + 210\ x\^7 - 5\ x\^8 - 10\ x\^9 + x\^10)\)\^2\ \((\(-1\) - 5\ x + 20\ x\^2 + 55\ x\^3 - 20\ x\^4 - 99\ x\^5 - 35\ x\^6 + 40\ x\^7 + 35\ x\^8 + 10\ x\^9 + x\^10)\)\^2, {"gS[7,{{2,25,1},{1,25,1,-.5}}]"}}\)},{},
{  },{},{  },{},{  },{},{  },{},{  },{},{  },{},{  },{},{  },{},{  },{},{  },{},{  \({"7,72", \((\(-3\) + x)\)\ x\^4\ \((1 + 
                      x)\)\ \((2 + x)\)\^3\ \((\(-14\) - x + 
                      x\^2)\)\ \((\(-49\) + 2548\ x - 30968\ x\^2 - 96726\ x\^3 + 1823780\ x\^4 + 7000924\ x\^5 - 12902715\ x\^6 - 92270010\ x\^7 - 85347003\ x\^8 + 226513490\ x\^9 + 487752048\ x\^10 + 47188638\ x\^11 - 671479754\ x\^12 - 585268936\ x\^13 + 149485599\ x\^14 + 469865070\ x\^15 + 190213272\ x\^16 - 80446972\ x\^17 - 85680334\ x\^18 - 11386546\ x\^19 + 11697400\ x\^20 + 4467526\ x\^21 - 357467\ x\^22 - 451356\ x\^23 - 40280\ x\^24 + 19538\ x\^25 + 3490\ x\^26 - 360\ x\^27 - 99\ x\^28 + 2\ x\^29 + x\^30)\)\^2, {"gS[7,{{.3,1,0,3.14},{4,7,1,-1.25},{3,28,1},{2,28,1,1.47},{1,7,1,2.9},{.3,1}}]"}}\)}};

8 regular

x =. ;graph8Data={{},{},{},{},{},{},{},
{},{  {"8,9 362880", -(((1 + x))^8), {{{2, 3, 
                        4, -3, -2}, 9}}}},{},{  \({"8,11,22", \(-\((1 - 2\ x - 5\ x\^2 + 2\ x\^3 + 4\ x\^4 + x\^5)\)\^2\), {{{4, 4, 3, 5, \(-3\), \(-4\), \(-4\)}, 11}}}\)},{  \({"8,12,1,10368", \((\(-2\) + x)\)\^2\ \((1 + x)\)\^8\ \((4 + 
                      x)\), {"gS[8,{{1,12}},{{3,-4,5,4,-3},12}]"}}, 
{"8,12,2,24", 
                x\^3\ \((1 + x)\)\^2\ \((3 + x)\)\^2\ \((\(-3\) + x\^2)\)\^2, {{{3, \(-4\), 2, 4, \(-3\)}, 12}}}\)},{  \({"8,13,PP3", \(-x\^4\)\ \((\(-2\) + x\^2)\)\^2\ \((\(-2\) + 2\ x + x\^2)\)\ \((6 + 6\ x + 
                      x\^2)\), {"PPn[3,{{.2,.3}}]"}}\)},{  },{  \({"8,15,30", \(-\((1 + x)\)\^2\)\ \((\(-1\) - x + x\^2)\)\^2\ \((\(-29\) - 31\ x - 4\ x\^2 + 4\ x\^3 + x\^4)\)\^2, {{{\(-6\), 
                      7, \(-5\), 9, 5, \(-7\), 6}, 156}}}\)},{  \({"8,16,32", \((2 + x)\)\^2\ \((4 + 
                      x)\)\ \((\(-2\) + x\^2)\)\^2\ \((\(-2\) - 8\ x - 8\ x\^2 + x\^4)\)\^2, {{{\(-6\), 7, \(-5\), 9, 5, \(-7\), 6}, 16}}}\)},{},{  \({"8,18,hy", \((\(-1\) + x)\)\^8\ \((1 + x)\)\^8\ \((8 + x)\), {{{3, \(-5\), 7, 5, \(-3\)}, 18}}}\)},{},
{   \({"8,20,hy", \((\(-2\) + x)\)\^4\ x\^10\ \((2 + x)\)\^4\ \((8 + x)\), {{{3, 7, 9}, 20}}}\)}};

allGraphs Development

Objective, Strategy.

This attempts to create all N-regular graphs having a specified skeleton. It creates a cycle of v vertices, with an optional inner cycle having symmetrical links to the outer circuit.The chords specified in n0 (using any of the croslnk formats) are then added. If this does not give a regular graph, infill is called with arguments vertx and done, an (initially empty) list of added edges. A list new of acceptable chords is created. These are sequentially added to the partly filled vertx, and infill called recursively. Backtracking occurs if no chord can be added, or if the graph is complete. Complete graphs are compared (using characteristic polynomials) with known graphs. For unknown graphs, the characteristic polynomial and the incantation are stored. To find the number of graphs (rather than unknown graphs) the parameter pr can be set to 1, whereupon all regular graphs are stored.
	The search space is reduced by initially specifying one of the chords from vertex 1. This need not go beyond the diameter, by symmetry. The attempt to shorten the search by specifying a maximum lengths is probably a waste of time.

Colour coding:-
green - module names, calls, and returns, red - error failures, blue - diagnostic print messages. pr is a global variable that controls the amount of diagnostic output, with graphs being shown as they are developed if pr>2. (It is True or False when creating the embeddings in Section 7; all output is suppressed in the False case.) If pr is 1, all graphs go into results; otherwise results only contains graphs that are NOT in the database.

Strategy:- 
allGraphs. Set up global variables - v,nv = no. of vertices, results, incant (incantations that will create each new graph), mx (the maximum number of edge ends at each vertex), depth (a diagnostic indication of the recursion depth), and kapol (a list of known characteristic polynomials).
Create the prescribed tst skeleton - an outer circuit, an optional inner circuit linked to the outer circuit, and the specified chords (usually defined by the extended LCF formulation, though others can be used). This creates vertx, the lists of vertices connected to each vertex.
If vertx is complete, call testGraph; otherwise call infill to find the sets of chords that create complete graphs, as follows:-
Do  1≤i≤nv-short;
  If vertx[[i]] incomplete, Do i+short≤j≤nv 
     If vertx[[j]] incomplete & chord not too long, add the edge and skip out of the Do loops. 
  Test any complete graph, backtrack if the Do loops have not found a new edge, or call infill recursively. vertx, done & comb1 have to be restored in this case.
Complete graphs are tested in testGraph; unknown graphs are stored in results and their incantations in incant; their characteristic polynomials will be added to kapol. If pr=1, all graphs go into results, to provide a count of the graphs with the specified structure. This will not be complete for multi-circuit graphs, as different interconnections are not explored.
On returning to allGraphs, the number of new graphs is reported. They can be viewed by ShowGraph[results[[n]]]; or ShowGraphArray[Partition[results,?]]; and added (manually) to graphNData.

allGraphs.

allGraphs[ed_, h_, n1_:  0, 
            n0_: {{0}, 1}, longest_:  10, shortest_:  2]:=Module[{r = 0, v, x = 0, inner = Abs[n1]},If[! IntegerQ[inner], 
                    Print["Faulty inner"]; Abort[]];v = h + inner;If[OddQ[v] && OddQ[ed], 
                    Print["Faulty Data, v & ed both Odd"]; Abort[]];
If[inner > 1, r = 1/5; 
                  x = -3.2 /inner];
  long = 
                  Min[longest, Floor[h/2] - 1];short = shortest;
results = {};incant = {};nv = v;mx = ed;depth = 0;   x =. ;kapol = ToExpression["graph" <> 
                        ToString[ed] <> "Data"][[v]];kapol = 
                  If[kapol == {}, {{}}, Transpose[kapol][[2]] ];
  
circuits = 
                  If[inner == 
                      0, {{1, h}}, {{1, h}, {r, inner, 1, 4.2  mx/h, 
                        Floor[h - 1/2 - mx/2]}}];
tst = gS[ed, circuits, n0];
If[pr > 1, Print["Starting graph"]; 
                    ShowLabeledGraph[tst]];
If[MatrixQ[vertx] && 
                      n0[[1]] ≠ {0}  ,
testGraph[tst = Graph[comb1, comb2]],If[!infill[vertx, {}],If[pr === True, Print["Fail"]]]];
If[pr === True || pr > 0,Print[{Length[
                            results], "New Graphs found."}]];];

infill.

infill[ver_, dn_]:=Module[{done = dn, i = 1, j, k, new = {}, oldv = ver, 
              oldcom = comb1},depth++;If[depth > 20, Print["Depth>20"]; Abort[]];Do[If[(i ≤ 
                            nv - short) && (Length[ver[[i]]] < 
                            mx),j = i + short;
Do[If[Length[ver[[j]]] < 
                                    mx && (! MemberQ[vertx[[j]], 
                                        i]) && ! MemberQ[
                                      done, {i, j}],If[pr > 3, 
                                      Print[{"     Added edge", i, j}]];
  AppendTo[vertx[[i]], j];AppendTo[
                                    vertx[[
                                      j]], i];AppendTo[comb1, {{i, j}}];AppendTo[done, {i, j}];If[Dimensions[vertx] == {nv, 
                                        mx},testGraph[
                                        tst = Graph[comb1, comb2]];If[pr > 3, 
                                        Print[{"infill graph at depth",  
                                        depth}]];Return[],infill[vertx, done];vertx = oldv;comb1 = oldcom;depth--;If[pr > 3, 
                                        Print[{"infill backtrack at \
depth" , depth}]]]],
{j, i + short, nv}]],
{i, nv - short}];If[pr > 3, 
                  Print[{"infill end at depth",  depth}]]];

testGraph.

testGraph. Find the characteristic polynomial. If it is not in kapol it is an unknown graph, so add to results, incant, & kapol before returning. (Store if pr=1 and graph is known.)

testGraph[gr_]:=Module[{kp, vy, w, y},x =. ;kp = 
                Factor[CharacteristicPolynomial[ToAdjacencyMatrix[gr], 
                      x]/(x - mx)];
If[MemberQ[kapol, kp],If[pr > 1, Print["Known Graph ", kp]];If[pr == 1,AppendTo[results, gr]];Return[True]];
  testv = vertx;
AppendTo[kapol, kp];If[pr > 1, Print[{"NEW", kp}]];AppendTo[results, gr];
  
vy = 
                Flatten[Drop[Transpose[vertx], 2]];y =. ;
Do[
                vy[[i]] = Max[0, vy[[i]] - Mod[i, nv]], {i, 
                  Length[vy]}];
While[
                Last[vy] ≠ 0 && Last[Most[vy]] == 0, 
                vy = RotateRight[vy]];
While[Last[vy] == 0, 
                vy = Most[vy]];v =. ;w =. ;z =. ;AppendTo[incant, 
                Hold[gS[w, z, {y, 1}]] /. {v -> nv, w -> mx, 
                    z -> circuits, y -> vy}];
If[pr > 2, ShowLabeledGraph[gr]];False];

testGraph[gr_]:=Module[{kp, l = circuits[[1, 2]], vy, w, y},x =. ;kp = 
                Factor[CharacteristicPolynomial[ToAdjacencyMatrix[gr], 
                      x]/(x - mx)];
If[MemberQ[kapol, kp],If[pr > 1, Print["Known Graph ", kp]];If[pr == 1,AppendTo[results, gr]];Return[True]];
  
AppendTo[kapol, kp];If[pr > 1, Print[{"NEW ", kp}]];AppendTo[results, gr];
  
vy = 
                Take[Flatten[Drop[Transpose[vertx], 2]], l];
If[l == nv, 
                Do[vy[[i]] = Max[0, vy[[i]] - Mod[i, l]], {i, 
                    l}]; 
While[
                  Last[vy] ≠ 0 && Last[Most[vy]] == 0, 
                  vy = RotateRight[vy]]; 
While[
                  Last[vy] == 0, vy = Most[vy]]];
v =. ;w =. ;z =. ;AppendTo[incant, 
                Hold[gS[w, z, {y, 1}]] /. {v -> nv, w -> mx, 
                    z -> circuits, y -> vy}];
If[pr > 2, ShowLabeledGraph[gr]];False];

subGraph Development

As vertex selection/deletion (card removal in [5]) reduce Hoffman-Singleton to interesting graphs, a selection/deletion procedure, subGraph, is provided, to create subgraphs from a sub-set of vertices. In many cases the sub-graph is regular with a lower number of edges per vertex. In others, some neighbours of deleted vertices may need to be joined to maintain regularity. This is done by inspection; programming a symmetrical join could be difficult.
	Strategy - create a known labelled graph as tst. Call subGraph with a list of vertices and a depth - the depth to which their neighbours are to be found. If depth is negative, these are to be deleted. If depth is positive, they are to be selected, deleting all others. The final deletion list is used to give nc1, nc2 for the revised graph by selecting edges from comb1 and vertices from comb2. The edges in inlist (using the vertex numbers of the original graph) are then added. The number of edges per vertex is calculated, and the result is stored as tst1 and plotted unles depth is real. pr>0 gives a list of the selections and deletions. 
 (Still needs input data validation.)

Clear[subGraph];tst =. ;tst1 =. ;subGraph[vlist_, depth_:  0, 
          inlist_: {}, graph_:  tst]:=Module[{g = graph, in, r, rul = {}, rc2 = Range[Length[comb2]], 
              s = vlist, si, t, n = {}, dp = {}, l = Length[vlist]},nc1 = Flatten[comb1, 1] ;nc2 = {} ;
Do[
                r = {};   
Do[
                  AppendTo[
                    r, Transpose[
                        Position[comb1, s[[j]] ]][[1]] ], {j, 
                    Length[s]}]; 
                t = {}; 
Do[
                  AppendTo[t, comb1[[r[[j]]]]], {j, Length[r]}]; 
                s = Union[Flatten[t]], {i, Abs[depth]}];
If[depth ≤ 0, r = s; 
                s = Complement[rc2, s], r = Complement[rc2, s]];
  
Do[
                AppendTo[rul, s[[i]] -> i], {i, Length[s]}];
in = inlist /. rul;If[pr > 0, 
                  Print[{"reject", r, "\nretain", s, "\n  as  ", 
                      s /. rul, "\nadd", in}]];
  
Do[
                If [! MemberQ[r, i], AppendTo[nc2, comb2[[i]]]], {i, 
                  Length[comb2]}];
  
Do[
                AppendTo[ 
                  dp, Transpose[Position[nc1, r[[i]]]][[1]]], {i, 
                  Length[r]}];dp = Union[Flatten[dp]];nc1 = {};  
Do[
                If [! MemberQ[dp, i], 
                  AppendTo[nc1, comb1[[i]]]], {i, Length[comb1]}];nc1 = nc1 /. rul;
  
Do[
                AppendTo[nc1, {in[[i]]}], {i, Length[in]}];tst1 = Graph[nc1, nc2];mx = 2  Length[tst1[[1]]]/Length[tst1[[2]]];If[! IntegerQ[mx], 
                Print[{"Not Regular, edges=", 
                    Length[tst1[[1]]], "vertices=", 
                    Length[tst1[[2]]]}]];
If[IntegerQ[depth], 
                ShowLabeledGraph[tst1]];tst1];

I had to write a clumsy search procedure to find 15 unconnected vertices for Ex.5 (my Ex.18. below), and have not attempted to find the sets needed for E.P.'s earlier examples. Several search attempts finally gave:-

regGraphId[graphNo[7, 50, 1], True];

tt = 
      Table[{i, vertx[[i]]}, {i, 
          50}];   
Do[
      n = Random[Integer, 50]; 
If[tt[[n, 1]] > 0, 
        k = tt[[n, 2]]; Do[tt[[k[[j]], 1]] -= 99, {j, 7}]], {m, 
        22}]; res = {}; 
    Do[If[tt[[i, 1]] > 0, AppendTo[res, tt[[i]]]], {i, Length[tt]}]; 
    res // tf
Length[res]

res=(({{2, {1, 3, 7, 45, 15, 26, 36}}, {4, {3, 5, 8, 27, 49, 19, 37}}, {10, {9, 11, 3, 6, 28, 38, 41}}, {13, {12, 14, 7, 27, 31, 41, 20}}, {16, {15, 17, 8, 28, 32, 42, 20}}, {18, {17, 19, 40, 50, 6, 14, 26}}, {23, {22, 24, 41, 37, 1, 30, 17}}, {25, {24, 26, 12, 9, 21, 32, 49}}, {29, {28, 30, 14, 5, 34, 45, 21}}, {33, {32, 34, 11, 40, 27, 47, 1}}, {35, {34, 36, 19, 22, 31, 42, 9}}, {39, {38, 40, 15, 44, 5, 31, 24}}, {43, {42, 44, 50, 21, 37, 7, 11}}, {46, {45, 47, 22, 50, 38, 8, 12}}, {48, {47, 49, 20, 44, 6, 30, 36}}}));

Confirm that these vertices are unconnected:-

rt = Transpose[res][[1]]
nn = Flatten[Transpose[res][[2]]];
Table[Position[nn, rt[[i]]], {i, Length[rt]}]

nnn = Union[nn]

rt

Find the vertices connected to 2:-

pr = 1;subGraph[{2}, 1. , {}];

Prime Projective Planes represented as regular graphs, PPn.

This creates projective planes with prime order n (powers >1 of primes are not covered), and represents them as 2n+2-regular graphs with each "Line" being a polygon of edges of a single colour. A test checks that every edge is incident in exactly one line. A set of colours (omitting very dark and very light colours) is defined. Point colouring needs more work.

SetOptions[ShowGraph, VertexStyle -> Disk[ .06], 
        VertexNumberPosition -> { .02, 0},  EdgeColor -> Black, 
        PlotRange -> All, EdgeStyle -> Normal];
\(\(colo = 
        ToExpression[{"Red", "Yellow", "Green", "Blue", "Indigo", "Violet", "Magenta", "Purple", "Turquoise", "Apricot", "Aquamarine", "AureolineYellow", "Banana", "Beige", "Bisque", "BlueViolet", "Brown", "BrownOchre", "Burlywood", "BurntSienna", "BurntUmber", "Brick", "BrownMadder", "CadetBlue", "CadmiumLemon", "CadmiumOrange", "CadmiumYellow", "Carrot", "Cerulean", "Chartreuse", "Chocolate", "ChromeOxideGreen", "CinnabarGreen", "Cobalt", "CobaltGreen", "Coral", "CornflowerBlue", "Cornsilk", "Cyan", "DarkGoldenrod", "DarkGreen", "DarkKhaki", "DarkOliveGreen", "DarkOrange", "DarkOrchid", "DarkSeaGreen", "DarkSlateGray", "DarkTurquoise", "DeepCadmiumRed", "DeepCobaltViolet", "DeepMadderLake", "DeepNaplesYellow", "DeepOchre", "DeepPink", "DeepSkyBlue", "DodgerBlue", "EmeraldGreen", "EnglishRed", "Firebrick", "ForestGreen", "Gainsboro", "GeraniumLake", "Gold", "Goldenrod", "GoldOchre", "Gray", "GreenishUmber", "Honeydew", "HotPink", "IndianRed", "Khaki", "LawnGreen", "ManganeseBlue", "Maroon", "MarsOrange", "MarsYellow", "MediumAquamarine", "MediumBlue", "MediumOrchid", "MediumPurple", "MediumSeaGreen", "MediumSlateBlue", "MediumSpringGreen", "MediumTurquoise", "MediumVioletRed", "Melon", "MidnightBlue", "Mint", "MintCream", "Moccasin", "Navajo", "Navy", "Oak", "Olive", "Orange", "OliveDrab", "Orchid", "PaleGoldenrod", "PaleGreen", "PaleTurquoise", "PaleVioletRed", "PapayaWhip", "Peach", "PeachPuff", "Peacock", "PermanentGreen", "PermanentRedViolet", "Peru", "Pink", "Plum", "PowderBlue", "PrussianBlue", "Raspberry", "RawSienna", "RawUmber", "RoseMadder", "RosyBrown", "RoyalBlue", "SaddleBrown", "Salmon", "SandyBrown", "SapGreen", "SeaGreen", "Sepia", "Sienna", "SkyBlue", "SlateBlue", "SlateGray", "Smoke", "SpringGreen", "SteelBlue", "TerreVerte", "Thistle", "Titanium", "Tomato", "TurquoiseBlue", "Ultramarine", "UltramarineViolet", "VanDykeBrown", "VenetianRed", "WarmGray", "Wheat", "YellowBrown", "YellowOchre"}];\)\)

Clear[ProjectivePlaneN, PPn, PPtest, pr];ProjectivePlaneN[any___] := PPn[any];PPn[n_, cntr_: {{0, 0}}, f_:  1]:=(*Create PPn Table & Graph*)
Module[{angl = If[EvenQ[n], 0, π/(n + 1), 0], c, cl, , 
                ix, iy, i0, j, jx, jy, j0, k, lines = {}, m = n n + n + 1, 
                n1 = n + 1, p, points = {cntr}, v, 
                theta = 2.  π/(n + 1)},
c = Range[m];pp = Table[0, {i, m}, {j, m}]  ;pl = {}  ;
  
Do[j = Mod[(i - 2), n] + 1; 
                  k = Floor[(i - 2)/n]; 
                  AppendTo[
                    points, {{Chop[(f^j) j Sin[k theta + angl]], 
                        Chop[(f^j) j Cos[k theta + angl]]}}], {i, 2, 
                    m}];
  pp[[1, 1]] = 1;
Do[pp[[j, 1]] = 1; 
                  pp[[1, j]] = j; 
                  Do[pp[[i + 1, i n + j]] = i n + j; 
                    pp[[i n + j, i + 1]] = i + 1, {i, n}], {j, 2, n1}];  
ix = 0;Do[i0 = n i + 1;iy = 0;
Do[j0 = n j + 1 ;   
Do[v=j0 + Mod[l + ix iy, n, 1];pp[[i0 + Mod[l, n, 1], v]] = 
                                      v,{l, n}];
iy++,{j, n}];
ix++,{i, n}];
  
If[pr > 1, Print[pp // tf]];If[pr < 0, pp = ppTest; Print["Test"];];
  
Do[p = Rest[Union[pp[[i]]]];AppendTo[pl, p];AppendTo[p, First[p]];
k = 1;While[k <= 
                            Length[p] && ! MemberQ[c, 
                              p[[k]]], k++;  ];
If[Length[p] >= k, 
                        cl = p[[k]]; 
                        c = Drop[c, Position[c, cl][[1]]], 
                        cl = First[c]; c = Rest[c]];AppendTo[points[[cl]], 
                        VertexColor -> (v = 
                              colo[[Mod[cl, Length[colo], 1]]])];
 (*Draw the line through these points*) 
Do[
                        AppendTo[
                          lines, {{p[[j]], p[[j + 1]]}, 
                            EdgeColor -> v}], 
{j, 
                          n1}],
{i, m}];
  Graph[lines, points]];

pl =. ;Clear[Testpp];Testpp:=Module[{OK = "OK", p, pl},pl = {};Do[p = Rest[Union[pp[[i]]]]; 
                  AppendTo[pl, p], {i, lp = Length[pp]}];  Do[  
Do[  
If[Length[
                                Intersection[pl[[i]], pl[[j]]]] > 1,Print[{"Fail", i, pl[[i]], 
                                    j, pl[[j]], 
                                    Intersection[pl[[i]], 
                                      pl[[j]]]}];If[pr ⩽ 0,OK = Fail,Abort[],]],{j, i + 1, lp}]
,{i, l = Length[pl] - 1}];Print[{OK}]];

Clear[gShort]; 
    gShort[g_] := 
      Module[{}, 
        gsh = Graph[Table[{g[[1, i, 1]]}, {i, Length[g[[1]]]}], 
            Table[{g[[2, j, 1]]}, {j, Length[g[[2]]]}]]; 
        toVertx[gsh]; ed = Length[vertx[[1]]]; mx = ed];

6. Tutorial.

Pre-defined Graphs, Combinatorica Instructions.

The graphKData files contain incantations for over 820 graph embeddings, which can be shown via Combinatorica instructions and graphNo[K:3, v:6, counter:1, incantation:1]; K is the number of edges per vertex (3 to 7), v is the number of vertices, counter is the position in the specified {K,v} list (which is ordered by decreasing symmetry), and incantation selects from several embeddings (if present).

\(\(\( (*Example\ 0\ *) \)\(Print[{"Data for the two 3-regular 6-vertex graphs:-
", 
graph3Data[\([6]\)]}];  (*First\ Graph*) 
      ShowLabeledGraph[tst1 = graphNo[3, 6, 1]];  (*Second\ Graph, \ 
      1  st\ embedding*) ShowGraph[
      tst2 = graphNo[3, 6, 2, 1]];  (*Second\ Graph, \ 2  nd\ embedding*) 
      ShowGraph[tst3 = graphNo[3, 6, 2, 2]];  (*Second\ Graph, \ 
      3  rd\ embedding, new\ options*) ShowLabeledGraph[
      tst4 = graphNo[3, 6, 2, 
            3] /. 
{regGraphOptions -> {EdgeColor -> Blue, VertexNumberColor -> Red, 
                VertexNumberPosition -> { .02, 0}, \ 
                VertexColor -> Black, VertexStyle -> Disk[ .07], 
                PlotLabel -> "3,6,2,3"}}];

    \(ShowGraphArray[{tst1, tst2, tst3, tst4}];\)\)\)\)

Session changes can be made to graph appearences by e.g.

regGraphOptions = {EdgeColor -> Blue, 
          VertexNumberColor -> Red, 
          VertexNumberPosition -> { .02, 0},  
          VertexColor -> Black, 
          VertexStyle -> Disk[ .07]};

Graph creation.

graphSynthesis[]  uses the default parameters ed_:3, c_:{{.5,5,2},{1,5}}, croslnk_:{} to draw the Petersen graph as a 2-stellated circuit (radius .5, vertices 1 to 5) within a normal circuit (radius 1, vertices 6 to 10). Symmetric edges (1-6, 2-7 etc) are automatically added between the circuits; three edges end on each vertex and so no extra cross-links are needed:-

ShowLabeledGraph[
        graphSynthesis[]];

Graph properties can be changed by changing the regGraphOptions:-

ShowLabeledGraph[
        graphSynthesis[] /. {regGraphOptions -> {EdgeColor -> Blue, 
                VertexNumberColor -> White, 
                VertexNumberPosition -> { .02, 0},  
                VertexColor -> Black, 
                VertexStyle -> Disk[ .07]}}];

The number of edges per vertex is given by ed_:3. The inner circuit is defined by {.5,5,2} as having 5 vertices on a circuit of radius 0.5, whilst it is 2-stellated by joining each vertex to the next but one. The outer circuit is defined by {1.,5} as 5 vertices on radius 1; it is joined by lines because the stellation number is 1 by default; it links symmetrically to the previous circuit. No cross-links are needed to make the graph regular.
	Now make some changes. The default options apply automatically. Changing ed to 4 gives 4 edges per vertex; putting {1,5} first (instead of second) makes vertices 1-5 the outer circuit. Pairs of links are automatically created between the circuits to give a regular 4th order graph. The inner circuit is rotated by -π/5 to give a pretty picture.

newgraph = 
        graphSynthesis[
          4, {{1, 5}, { .5, 5, 3, -π/5. }}];

    ShowLabeledGraph[newgraph];

Next, draw a nonagon as the single circuit {{1.,9}} and introduce crosslinks in the form {{1,3,-1}}. This inserts an edge of length 3, starting at vertex 1 and ending at vertex 4 (though it is actually "undirected"). It then steps by -1 (i.e. back from vertex 4 to vertex 3) and inserts another edge of length 3, to vertex 6. This process is repeated, stopping when it reaches a full vertex (in this case it when cannot insert an edge from vertex 7 to vertex 1). Three vertices are reported as incomplete. (Nonagons cannot be 3-regular.)

pr = 3; 
    ShowLabeledGraph[
      graphSynthesis[3, {{1, 9}}, {{1, 3, -1}}]];

Change the order to 4, and a complete 4-regular nonagon is now drawn and can be investigated. It is the first 4-regular graph in the data, and has 18 automorphisms:-

\(\(\( (*Example\ 3  a*) \)\(nonagon4 = 
      graphSynthesis[4, {{1. , 9}}, {{1, 3, \(-1\)}}]; 
    ShowLabeledGraph[nonagon4]; regGraphId[nonagon4]

    Automorphisms[nonagon4]\)\)\)

The data entry shows that it is the first entry in the graph4Data list, having 18 automorphisms and a CharacteristicPolynomial with 5 factors (the (x-4) factor of a 4-regular graph is implicit). The incantation is an LCF4 notation {(3^9)} :-

graph4Data[[9, 3]]

Any graph in the database can be created by graphNo[ edges, vertices, index, {isomorph}] and shown by ShowGraph or ShowLabeledGraph; if it is given a temporary name such as tst its characteristic polynomial can be checked:-

ShowLabeledGraph[tst = graphNo[4, 9, 3]]; 
    cp[tst]

Different Graph embeddings and incantations.

Many graphs can be drawn as different embeddings and in different ways. This is demonstrated with the Moebius-Kantor graph, which can be drawn as a 3-stellated hexagon inside a normal hexagon, or as a hexadecagon with a croslnk edge length of 5 and a step of 1. It can also be created as a generalized Petersen graph, and by using the LCF notation {{5,-5},6} where the 6 is the "power". This symmetric cubic graph is remarkable for its number of automorphisms. As the specific automorphisms are rarely of interest, the instruction Length[Automorphisms[..]] is useful to indicate how many symmetries the graph possesses. (This value is used to order the entries in the data.). The calculation is slow in this example, and so has been commented out.  The first and third incantations give the same embedding, but the inner stellation is slightly rotated in the genPetersen case, as many such graphs would otherwise have coincident edges.

MK = {MK1 = 
          graphSynthesis[
            3, {{1. , 8}, { .5, 8, 3}}], 
MK2 = 
          graphSynthesis[
            3, {{1. , 16}}, {{1, 5, 1}}], 
MKp = 
          genPetersen[8, 3], 
        MKlcf = graphSynthesis[3, {{1. , 16}}, {{5, -5}, 6}]}; 
    ShowGraphArray[MK];

A graph may have several visually distinct embeddings (not counting rotations and reflections) as above. They were originally tested for isomorphism by the local version of Eric Weisstein's MyIsomorphicQ from his MathWorld Graph package, but this proved inaccurate. It also involved tests that were irrelevant for regular graphs, and so has been replaced by a new test that compares CharacteristicPolynomials :-

{regIsomorphicQ[MK1, MK2], 
      regIsomorphicQ[MK1, MKp], regIsomorphicQ[MK1, MKlcf]}

allGraph demonstration.

allGraphs is an experimental procedure that attempts to create and identify graphs fitting a specification. If pr=1, all graphs are found, otherwise only unknown graphs are found; they are put into results. Example 6a finds all 8/4 cubical graphs with a diametral edge.

pr = 1;tst=allGraphs[3, 8, 0, {{0, 4}, 1}];
ShowGraphArray[results];

Example 6b starts with circuits of 12 vertices and an edge (1,3); 8 new graphs are found. Tests show that they all have Hamiltonian cycles.

pr = 0;allGraphs[3, 12, 4, {{2}, 1}];
ShowGraphArray[results];

i = 1; incant[[i]]
toVertx[results[[i]]]; hamiltonianCycle[]

Generalized LCF notation.

LCF notation can be found for graphs with some symmetry. A first choice can often be improved by changing a step so that the first incomplete vertex is filled. Thus the -5 in position 2 in the next example is changed to +3 to fill the incomplete vertex 5 (with a symmetrical change of the penultimate +5), and the resulting regular graph has 12 automorphisms.

pr = 1; 
    ShowLabeledGraph[
      graphSynthesis[
        3, {{1. , 18}}, {{5, -5, 7, -7, 5, -5}, 
          3}]];

\(\(\( (*Example\ 7  b*) \)\(\(ShowLabeledGraph[
        new18 = graphSynthesis[
              3, {{1. , 18}}, {{5, 3, 7, \(-7\), \(-3\), \(-5\)}, 
                3}] /. {regGraphOptions -> {PlotLabel -> "New18"}}];\)

    Length[Automorphisms[new18]]\)\)\)

The LCF procedure has been generalized to any value of ed and to only add edges if the target vertex is not full.
	Some 4th & higher order graphs have an "LCF4" etc. notation. The Meringer and Wong graphs were analysed by listing the edge lengths, noting the repetitions (Wong gives 6 {a,±b} sets) and putting these into the LCF format. The Hoffman-Singleton graph in [5] has 5 cycles of 10 vertices, each with 5 chords. Listing 40 chords in order, zeroing the last two, gave the incantation.

Any Hamiltonian graph can be drawn by the pseudoLCF notation, by listing the length of each edge in turn and a power 1.  Many graphs (with some symmetry) can be created using higher powers than 1, with a shorter list that may give an isomorph that looks different:-

(*Example 8*)
ShowGraphArray[{tst1=gS[3,{{1,12}},{{-5,5,-5,5,-5,5},1}],
tst2=gS[3,{{1,12}},{{-3,3,5,-5},2}],gS[3,{{1,12}},{{-5,5},3}]}];regIsomorphicQ[tst1, tst2]

A length of 0 can be used if a particular edge is already full. The "power" is 1 as the cycle is not repeated. Thus the Frucht graph is given by:-

ShowLabeledGraph[
        frLCF = graphSynthesis[
            3, {{1. , 12}}, {{-5, -2, -4, 2, 5, 0, 2}, 
              1}]];

    regGraphId[frLCF]

Asymmetric & other graphs.

Asymmetric graphs, such as the standard embedding of the Frucht graph, can be drawn by deleting and replacing edges of a related graph, using a negated vertex indices for the deleted edges. Example 10 sets up a starter for the Frucht graph:-

ShowLabeledGraph[
        graphSynthesis[
          3, {{1. , 7, 1, π/3.5}, { .6, 4, 0,  .67}, {0, 
              1}}]];

Deletions are described by two or 5 indices {-2,9} being needed here to remove a single line; simple insertions use two indices e.g. {2,8} whilst normal insertions use one index and two steps; {8,1,1} is used as an example to put one edge from 8 to 9 ({8,9} would do the same). The 5-element insertion/deletion option is often used for larger graphs.

ShowLabeledGraph[
        graphSynthesis[
          3, {{1. , 7, 1, π/3.5}, { .6, 4, 0,  .67}, {0, 1}}, {{-2, 
              9}, {2, 8}, {8, 1, 1}}]];

Multiple deletions use 5 numbers, as in the Chvatal embedding, Example 10b, where {-3,9,1,1,4} effects 4 deletions, starting with edge {3,9} and incrementing by {1,1} to delete {4,10}, {5,11}, & {6,12}:-

ShowLabeledGraph[
        gS[4, {{0.2, 2, 0}, { .6, 5, 0, - .315}, {1. , 
              5. , -1,  .3 3.1416}}, {{-1, 6, 1, 1, 2}, {-3, 9, 
              1, 1, 4}, {-7, 8}, {1, 6, 1}, {2, 3, 1}, {3, 5, 1}, {6, 1, 
              1}}]];

Hamiltonian cycles & near-Hamiltonian cycles.

hamiltonianCycle[n] operates on vertx provided by the current GraphSynthesis data or after regGraphId[tst,True] if the graph was generated by subGraphs or a Combinatorica procedure. It is a modification of the version in [4]. It finds the first n hamiltonian cycles for the most recent graph, or all the non-Hamiltonian sequences. These can then be shown via replot[n]. Two are shown for the Frucht graph, the first matching that in the earlier example:-

tst = graphNo[3, 12, -5]; 
    hamiltonianCycle[5]

    Length[vertx]

    ShowGraphArray[{tst, replot[1], replot[5]}];

Non-Hamiltonian graphs may have many near-hamiltonian cycles, which are also found by hamiltonianCycle[]. This creates the global variable Hc as a list of circuits even if the graph is not Hamiltonion. There are 600 such circuits for the Coxeter graph; a few are shown below (SLOW). (Only one of these is included in the database.)

ShowLabeledGraph[
      Coxeter = 
        gS[3, {{ .2, 7, 3,  .175}, {1. , 7, 0}, { .5, 7, 
              2,  .35}, { .7, -7, 1, - .44}}]]; 
    hamiltonianCycle[]

    ShowGraphArray[{replot[3], replot[132, -1.3], replot[144, 2], 
          replot[470,  .2]}];

Incantations can be synthesized for replots by setting up the appropriate circuits and adding the edges:-

ShowLabeledGraph[
        replot[3]];

ShowLabeledGraph[
      Coxeterc = 
        gS[3, {{1, 27}, {0, 1, 0, -1}}, {{1, 7}, {2, 11}, {3, 21}, {4, 
              17}, {5, 24}, {6, 14}, {8, 19}, {9, 23}, {10, 16}, {12, 
              25}, {13, 20}, {15, 28}, {18, 26}, {22, 28}, {27, 28}}]]; 
    regGraphId[Coxeterc]

Disconnected graphs.

Disconnected graphs can be created, but few are in the database (their names end in nx); they have no cycles, hamiltonian or otherwise:-

ShowLabeledGraph[
      tst = gS[3, {{1. , 8, 2}}, {{1, 4, -1}}]]; 
    regGraphId[tst]

    hamiltonianCycle[]

ShowGraph[
      tst = gS[3, {{1. , 3, 1,  .525}, {2.3, 12, 4,  .52}, {1. , 3, 1, 
              2.1}}, {{6, 5, -1}}]]; regGraphId[tst]

    hamiltonianCycle[]

genPetersen graphs.

genPetersen[inner orbit size o, stellation s] creates a stellated inner orbit of the specified size o linked to an outer orbit of the same size, unless the stellation s is diametral, s=o/2, when the sign of the stellation determines how the graph is completed. If it is positive, the inner vertices link to two outer vertices, so that the outer orbit has 2o vertices. If it is negative, the inner orbit vertices are pairwise joined to give each vertex three edges. If o is negative and < -2, a variety of different graphs involving three orbits are given by some values of s:-

gp = {genPetersen[4, -2], 
        genPetersen[4, 2], genPetersen[5, 3], genPetersen[-3, 1], 
        genPetersen[-3, 3], genPetersen[6, 2], genPetersen[6, -3], 
        genPetersen[6, 3], genPetersen[-4, 1], 
        genPetersen[-4, -4], genPetersen[-5, 3], 
        genPetersen[-5, -5]}; 
    ShowGraphArray[Partition[gp, 6]];

Peninsula graphs.

Peninsula graphs (12/4/4) are graphs that are non-hamiltonian because they become disconnected on deletion of a "bridge edge". They may have several bridge edges. Single-bridge "islands" must have odd numbers of edges and nodes. They are stored at the end of the other graphs of the same size, as "Pvn". Only a few are stored; others can be built up with similar "islands" and "archipelagos".

ShowGraphArray[{{graphNo[3, 10, -1], 
            graphNo[3, 10, -1, 2], graphNo[3, 14, -1], 
            graphNo[3, 14, -2]}, {graphNo[3, 16, -1], 
            graphNo[5, 14, -1], 
            graphNo[5, 14, -1, 2]}}];

subGraph examples.

subGraph was written to mechanise the processes described by Ed Pegg Jnr. in [5], to give subgraphs of the Hoffman-Singleton graph. Some are shown here. More general applications of subGraph are in the next section. The graph is plotted if depth is an integer (i.e. not if it is real, with a decimal point).
I have not always found pretty plots, and so have put some new graphs in the database in subGraph form. Some include names in the Gallery.
(Examples 16 & 17 deleted.)

Some of Ed Pegg's examples & other HS subgroups.

I had to write a clumsy search procedure to find 15 unconnected vertices for Ex.5 (in subGraphs development).

tst =. ; tst1 =. ; HS = graphNo[7, 50, 1];

(*Example 18.  Ed Pegg[5 Ex .  4],  
      Selecting 15 unconnected vertices give (Subscript[O, 4 ])*) pr = 0; ShowGraph[
      tst = subGraph[{2, 4, 10, 13, 16, 18, 23, 25, 29, 33, 35, 39, 43, 46, 
            48}, 0. , {}, graphNo[7, 50, 1]]]; mx = 4; 
    regGraphId[tst, True]

hamiltonianCycle[
      100]  
Length[Hc]

nn = 90; 
    ShowGraphArray[{{replot[1 + nn], replot[2 + nn]}, {replot[3 + nn], 
          replot[4 + nn]}, {replot[5 + nn], replot[6 + nn]}, {replot[7 + nn], 
          replot[8 + nn]}, {replot[9 + nn], replot[10 + nn]}}];

The neighbours of vertex 2 are also deleted for the next example.

pr = 0;
subGraph[{1, 2, 3, 4, 7, 10, 13, 15, 16, 18, 23, 25, 26, 29, 33, 
              35, 36, 39, 43, 45, 46, 48}, 0. , {}, graphNo[7, 50, 1]];
regGraphId[tst1, True]

Using subGraph to delete any connected pair of Hoffman-Singleton vertices and their neighbours gives the Sylvester Graph:-

pr = 0;
subGraph[{1, 2}, -1. , {}, 
            graphNo[7, 50, 1]];
regGraphId[tst1, True]

pr = 0;subGraph[{1, 5, 23, 50}, -1. , {}, graphNo[7, 50, 1]];
regGraphId[tst1, True]
ShowLabeledGraph[
            graphNo[4, 24, 2]];

Deleting any HS single vertex and its neighbours gives a 42 vertex 6-regular graph, (Balaban 11 cage) for which I have not found a symmetrical embedding:-

pr = 0;  subGraph[{48}, -1, {}, 
        graphNo[7, 50, 1]];
regGraphId[tst1, True]

The next example eliminates 3 (selected) vertices and their neighbours. The graph is converted to a symmetric Hamiltonian cycle. (Several cycles were found, and the first symmetrical one put in database)

pr = 0;subGraph[{1, 2, 5}, -1. , {}, 
            graphNo[7, 50, 1]]
regGraphId[tst1, True]
hamiltonianCycle[3];
ShowLabeledGraph[replot[3]];

The next graph eliminates 4 (selected) vertices and their neighbours. A symmetrical embedding has been put in the database.

pr = 0;subGraph[{2, 4, 6, 9}, -1. , {}, graphNo[7, 50, 1]]
regGraphId[tst1, True]
ShowLabeledGraph[
            graphNo[4, 24, 2]];

A random search produced the Desargues graph. The 5 specified vertices each share one neighbour pairwise, so 30 are rejected:-

pr = 0;subGraph[{8, 13, 37, 45, 48}, -1. , {}, graphNo[7, 50, 1]]
regGraphId[tst1, True]
ShowLabeledGraph[
            graphNo[3, 20, 2]];

Deleting two unconnected vertices and their neighbours gives an irregular 35 vertex graph with 4 vertices having 6, the rest 5, edges:-

tst1=subGraph[{1, 10}, -1, {}, 
            graphNo[7, 50, 1]]
regGraphId[tst1, True]

None of the extra edges join pairs of vertices, so they cannot be cancelled.

Other subGraph examples.

These graphs were added to the database using subGraph incantations.
Dropping the central 4 of graphNo[4,24,1] ( the SmallRhombicuboctahedron, a nice example to work on) requires the circuit 13 to 20 to be joined. This can be done in several ways.

subGraph[{21, 22, 23, 
            24}, -1, {{5, 8}, {6, 11}, {7, 10}, {9, 12}}, 
          graphNo[4, 24, 1]]
regGraphId[tst1, True]

subGraph[{21, 22, 23, 
            24}, -1, {{5, 9}, {6, 10}, {7, 11}, {8, 12}}, 
          graphNo[4, 24, 1]]
regGraphId[tst1, True]

subGraph[{21, 22, 23, 
            24}, -1, {{5, 9}, {6, 10}, {7, 12}, {8, 11}}, 
          graphNo[4, 24, 1]]
regGraphId[tst1, True]

Removing the outer and inner squares and repairing symmetrically gave a highly symmetrical graph that was not originally in graph4Data.

subGraph[{1, 2, 3, 4, 21, 22, 23, 24}, 
            0, {{5, 8}, {6, 11}, {7, 10}, {9, 12}, {13, 16}, {14, 19}, {15, 
                18}, {17, 20}}, graphNo[4, 24, 1]];
regGraphId[tst1, True]

Removing four vertices from the first 4,16 graph proved that I had not found all the 12-element graphs:-

subGraph[{1, 5, 9, 13}, 0, {}, 
            graphNo[6, 16, 1]];
regGraphId[tst1, True]

PPn  ( n prime) examples.

pr = 0; ShowLabeledGraph[PP3 = PPn[3]]; Testpp

Fig 1. Demonstration of PP3 creation and testing.
{OK} means that all pairings of vertices occur in exactly one line.

pp // tf

Fig 2. pp for PP3. Each row is a linelist, with zeroes for absent edges.
Note the basic structure of the first 4 rows and columns.

Transpose[pl] // tf

Fig 3. pl for PP3. Each column is a linelist.

gShort provides the conversions (to gsh) that allow RegGraph instructions,

gShort[PPn[3, {{0,  .3}}]]; ShowLabeledGraph[gsh]; 
    regGraphId[gsh]

PP4, PP8, PP9.

Data, from http://www.uwyo.edu/moorhouse/pub/planes/pg24.txt, pg28.txt, pg29.txt, has been converted to PPn format in pp4-8-9.nb. The results are duplicated here as pl4, pl8, pl9  and demonstrated by converting to ppTest, plotting and validating. I am attempting to extend the programme to reproduce these planes.

pl =. ; ppTest =. ; 
    plTopp[pl_] := 
      Module[{m = Length[pl], k}, ppTest = Table[0, {i, m}, {j, m}]; 
        Do[k = pl[[j]]; 
          Do[ppTest[[j, k[[i]]]] = k[[i]], {i, Length[k]}], {j, 
            m}]]

pl4=Transpose[(({{1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5}, {2, 6, 10, 14, 18, 6, 7, 8, 9, 6, 7, 8, 9, 6, 7, 8, 9, 6, 7, 8, 9}, {3, 7, 11, 15, 19, 10, 11, 12, 13, 11, 10, 13, 12, 13, 12, 11, 10, 12, 13, 10, 11}, {4, 8, 12, 16, 20, 14, 16, 15, 17, 17, 15, 16, 14, 15, 17, 14, 16, 16, 14, 17, 15}, {5, 9, 13, 17, 21, 18, 20, 21, 19, 21, 19, 18, 20, 20, 18, 19, 21, 19, 21, 20, 18}}))];

plTopp[pl4];pr=-1;ShowLabeledGraph[PPn[4],VertexStyle->Disk[.04]];Testpp

pl8=(({{1, 2, 3, 4, 5, 6, 7, 8, 9}, {1, 10, 11, 12, 13, 14, 15, 16, 17}, {1, 18, 19, 20, 21, 22, 23, 24, 25}, {1, 26, 27, 28, 29, 30, 31, 32, 33}, {1, 34, 35, 36, 37, 38, 39, 40, 41}, {1, 42, 43, 44, 45, 46, 47, 48, 49}, {1, 50, 51, 52, 53, 54, 55, 56, 57}, {1, 58, 59, 60, 61, 62, 63, 64, 65}, {1, 66, 67, 68, 69, 70, 71, 72, 73}, {2, 10, 18, 26, 34, 42, 50, 58, 66}, {2, 11, 19, 44, 36, 52, 28, 68, 60}, {2, 12, 59, 23, 45, 40, 55, 32, 72}, {2, 13, 35, 21, 29, 46, 54, 63, 71}, {2, 14, 51, 20, 69, 38, 47, 30, 64}, {2, 15, 67, 22, 53, 39, 48, 31, 65}, {2, 16, 43, 24, 61, 70, 41, 56, 33}, {2, 17, 27, 25, 37, 62, 49, 57, 73}, {3, 10, 19, 53, 37, 45, 29, 69, 61}, {4, 10, 43, 21, 68, 38, 57, 32, 65}, {5, 10, 35, 24, 52, 49, 31, 64, 72}, {6, 10, 59, 20, 28, 39, 46, 56, 73}, {7, 10, 67, 25, 44, 41, 55, 30, 63}, {8, 10, 27, 22, 60, 70, 40, 47, 54}, {9, 10, 51, 23, 36, 62, 48, 33, 71}, {3, 11, 18, 35, 43, 51, 27, 67, 59}, {3, 12, 42, 24, 60, 39, 57, 30, 71}, {3, 13, 26, 25, 36, 47, 56, 65, 72}, {3, 15, 50, 20, 68, 40, 49, 33, 63}, {3, 16, 58, 23, 44, 38, 54, 31, 73}, {3, 14, 66, 22, 52, 62, 41, 46, 32}, {3, 17, 34, 21, 28, 70, 48, 55, 64}, {4, 11, 66, 25, 45, 39, 54, 33, 64}, {6, 11, 26, 22, 61, 38, 49, 55, 71}, {8, 11, 58, 20, 29, 41, 48, 57, 72}, {9, 11, 34, 24, 53, 47, 32, 63, 73}, {7, 11, 42, 21, 69, 62, 40, 56, 31}, {5, 11, 50, 23, 37, 70, 46, 30, 65}, {4, 13, 18, 52, 61, 40, 48, 30, 73}, {9, 12, 18, 28, 69, 41, 49, 54, 65}, {8, 15, 18, 44, 37, 56, 32, 71, 64}, {7, 17, 18, 60, 53, 38, 46, 33, 72}, {6, 14, 18, 36, 45, 70, 57, 31, 63}, {5, 16, 18, 68, 29, 62, 39, 47, 55}, {4, 17, 50, 59, 19, 41, 47, 31, 71}, {5, 12, 66, 27, 19, 38, 48, 56, 63}, {9, 16, 26, 67, 19, 40, 46, 57, 64}, {8, 14, 42, 35, 19, 55, 33, 65, 73}, {6, 15, 34, 43, 19, 62, 54, 30, 72}, {7, 13, 58, 51, 19, 70, 39, 49, 32}, {4, 12, 26, 35, 20, 44, 53, 70, 62}, {4, 14, 34, 67, 23, 60, 29, 49, 56}, {4, 15, 58, 27, 24, 36, 69, 46, 55}, {4, 16, 42, 51, 22, 28, 37, 63, 72}, {6, 12, 58, 67, 21, 52, 37, 47, 33}, {7, 12, 50, 43, 22, 36, 29, 64, 73}, {8, 12, 34, 51, 25, 68, 61, 46, 31}, {7, 14, 26, 59, 24, 68, 37, 48, 54}, {8, 17, 26, 43, 23, 52, 69, 39, 63}, {5, 15, 26, 51, 21, 60, 45, 41, 73}, {6, 16, 50, 35, 25, 60, 69, 48, 32}, {7, 15, 66, 35, 23, 28, 61, 47, 57}, {9, 17, 58, 35, 22, 68, 45, 56, 30}, {7, 16, 34, 27, 20, 52, 45, 71, 65}, {9, 13, 66, 43, 20, 60, 37, 55, 31}, {5, 17, 42, 67, 20, 36, 61, 54, 32}, {6, 17, 66, 51, 24, 44, 29, 40, 65}, {9, 14, 50, 27, 21, 44, 61, 39, 72}, {5, 13, 34, 59, 22, 44, 69, 57, 33}, {6, 13, 42, 27, 23, 68, 53, 41, 64}, {8, 16, 66, 59, 21, 36, 53, 49, 30}, {5, 14, 58, 43, 25, 28, 53, 40, 71}, {8, 13, 50, 67, 24, 28, 45, 62, 38}, {9, 15, 42, 59, 25, 52, 29, 70, 38}}));

plTopp[pl8];pr=-1;ShowLabeledGraph[PPn[8],VertexStyle->Disk[.04],EdgeStyle->Thin];Testpp

This reorders pl so that the pl=Union[pl] test fails. It has been commented out of Testpp.

pl9=(({{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, {1, 11, 12, 13, 14, 15, 16, 17, 18, 19}, {1, 20, 21, 22, 23, 24, 25, 26, 27, 28}, {1, 29, 30, 31, 32, 33, 34, 35, 36, 37}, {1, 38, 39, 40, 41, 42, 43, 44, 45, 46}, {1, 47, 48, 49, 50, 51, 52, 53, 54, 55}, {1, 56, 57, 58, 59, 60, 61, 62, 63, 64}, {1, 65, 66, 67, 68, 69, 70, 71, 72, 73}, {1, 74, 75, 76, 77, 78, 79, 80, 81, 82}, {1, 83, 84, 85, 86, 87, 88, 89, 90, 91}, {2, 11, 20, 29, 38, 47, 56, 65, 74, 83}, {2, 12, 21, 31, 40, 67, 58, 76, 85, 49}, {2, 13, 30, 22, 42, 69, 78, 60, 51, 87}, {2, 14, 48, 23, 41, 32, 70, 62, 90, 82}, {2, 15, 75, 24, 68, 33, 43, 89, 63, 55}, {2, 16, 57, 25, 50, 34, 88, 44, 81, 73}, {2, 17, 66, 26, 86, 35, 52, 80, 45, 64}, {2, 18, 84, 27, 77, 36, 61, 71, 54, 46}, {2, 19, 39, 28, 59, 37, 79, 53, 72, 91}, {3, 11, 22, 31, 41, 68, 50, 86, 77, 59}, {3, 12, 20, 30, 39, 48, 75, 57, 66, 84}, {3, 13, 29, 21, 44, 62, 89, 80, 71, 53}, {3, 14, 65, 25, 85, 37, 78, 43, 54, 64}, {3, 15, 38, 27, 58, 35, 51, 70, 81, 91}, {3, 16, 83, 28, 76, 32, 69, 52, 63, 46}, {3, 17, 47, 24, 40, 36, 60, 88, 72, 82}, {3, 18, 56, 26, 49, 33, 42, 79, 90, 73}, {3, 19, 74, 23, 67, 34, 87, 61, 45, 55}, {4, 11, 30, 21, 43, 70, 88, 52, 61, 79}, {4, 12, 29, 22, 45, 90, 63, 81, 54, 72}, {4, 13, 20, 31, 46, 82, 55, 73, 64, 91}, {4, 14, 38, 84, 28, 49, 68, 34, 60, 80}, {4, 15, 65, 57, 26, 76, 41, 36, 87, 53}, {4, 16, 47, 75, 23, 58, 86, 37, 42, 71}, {4, 17, 83, 39, 27, 67, 50, 33, 78, 62}, {4, 18, 74, 48, 24, 85, 59, 35, 69, 44}, {4, 19, 56, 66, 25, 40, 77, 32, 51, 89}, {5, 11, 75, 26, 40, 34, 69, 62, 54, 91}, {5, 12, 65, 27, 59, 32, 42, 88, 80, 55}, {5, 13, 38, 57, 24, 67, 77, 37, 52, 90}, {5, 14, 74, 39, 21, 86, 36, 51, 63, 73}, {5, 15, 56, 30, 28, 85, 50, 71, 45, 82}, {5, 16, 20, 49, 41, 35, 78, 61, 89, 72}, {5, 17, 29, 48, 25, 58, 68, 87, 79, 46}, {5, 18, 83, 66, 23, 31, 60, 43, 53, 81}, {5, 19, 47, 84, 22, 76, 33, 70, 44, 64}, {6, 11, 84, 25, 67, 35, 42, 53, 63, 82}, {6, 12, 38, 23, 50, 36, 69, 79, 89, 64}, {6, 13, 65, 48, 28, 40, 86, 33, 61, 81}, {6, 14, 56, 75, 27, 31, 87, 52, 44, 72}, {6, 15, 74, 66, 22, 49, 37, 88, 62, 46}, {6, 16, 29, 39, 26, 85, 77, 60, 70, 55}, {6, 17, 20, 76, 59, 34, 51, 43, 71, 90}, {6, 18, 47, 57, 21, 68, 32, 78, 45, 91}, {6, 19, 83, 30, 24, 58, 41, 80, 54, 73}, {7, 11, 66, 28, 58, 36, 78, 44, 90, 55}, {7, 12, 74, 25, 41, 33, 60, 52, 71, 91}, {7, 13, 56, 39, 23, 76, 68, 35, 88, 54}, {7, 14, 47, 30, 26, 67, 59, 89, 81, 46}, {7, 15, 83, 48, 21, 77, 34, 42, 72, 64}, {7, 16, 65, 84, 24, 31, 51, 79, 62, 45}, {7, 17, 38, 75, 22, 85, 32, 61, 53, 73}, {7, 18, 20, 40, 50, 37, 87, 70, 80, 63}, {7, 19, 29, 57, 27, 49, 86, 69, 43, 82}, {8, 11, 48, 27, 76, 37, 60, 89, 45, 73}, {8, 12, 56, 24, 86, 34, 78, 70, 53, 46}, {8, 13, 74, 84, 26, 58, 50, 32, 43, 72}, {8, 14, 83, 57, 22, 40, 35, 79, 71, 55}, {8, 15, 47, 39, 25, 31, 69, 61, 80, 90}, {8, 16, 38, 66, 21, 59, 33, 87, 54, 82}, {8, 17, 65, 30, 23, 49, 77, 44, 63, 91}, {8, 18, 29, 75, 28, 67, 41, 51, 88, 64}, {8, 19, 20, 85, 68, 36, 42, 52, 62, 81}, {9, 11, 57, 23, 85, 33, 51, 80, 72, 46}, {9, 12, 47, 28, 77, 35, 87, 43, 62, 73}, {9, 13, 83, 75, 25, 49, 59, 36, 70, 45}, {9, 14, 29, 66, 24, 76, 50, 42, 61, 91}, {9, 15, 20, 67, 86, 32, 60, 79, 44, 54}, {9, 16, 74, 30, 27, 40, 68, 53, 90, 64}, {9, 17, 56, 84, 21, 41, 37, 69, 81, 55}, {9, 18, 65, 39, 22, 58, 34, 52, 89, 82}, {9, 19, 38, 48, 26, 31, 78, 88, 71, 63}, {10, 11, 39, 24, 49, 32, 87, 71, 81, 64}, {10, 12, 83, 26, 68, 37, 51, 61, 44, 82}, {10, 13, 47, 66, 27, 85, 41, 34, 79, 63}, {10, 14, 20, 58, 77, 33, 69, 88, 53, 45}, {10, 15, 29, 84, 23, 40, 59, 78, 52, 73}, {10, 16, 56, 48, 22, 67, 36, 43, 80, 91}, {10, 17, 74, 57, 28, 31, 42, 70, 89, 54}, {10, 18, 38, 30, 25, 76, 86, 62, 72, 55}, {10, 19, 65, 75, 21, 50, 35, 60, 90, 46}}));

plTopp[pl9];pr=-1;ShowLabeledGraph[PPn[9],VertexStyle->Disk[.04],EdgeStyle->Thin];Testpp

7. Small Graph Embeddings. "A Gallery of Graphs."

G3

G3 = {}; pr = False; grn = 0; grt = 0; 
    Do[col = Blue; 
      Do[Do[lbl = graph3Data[[ii, jj, 1]]; 
          d = graph3Data[[ii, jj, 3]]; If[pr, Print[{lbl, d}]]; 
          If[Head[d[[k]]] === String, 
            AppendTo[
              G3, {ToExpression[d[[k]]] /. 
                  regGraphOptions -> {PlotLabel -> lbl, 
                      EdgeColor -> col}}], 
            AppendTo[
              G3, {gS[3, {{1, ii}}, d[[k]]] /. 
                  regGraphOptions -> {PlotLabel -> lbl, 
                      EdgeColor -> col}}]]; grn++; 
          col = Black, {k, Length[graph3Data[[ii, jj, 3]]]}]; grt++; 
        col = Green, {jj, Length[graph3Data[[ii]]]}], {ii, 3, 
        Length[graph3Data]}]; {grn, grt, Length[G3]}

gallery[G3, 1, 5, 10];

gallery[G3, 51, 5, 10];

gallery[G3, 101, 5, 10];

gallery[G3, 151, 5, 10];

gallery[G3, 201, 5, 10];

gallery[G3, 251, 5, 10];

gallery[G3, 301, 5, 10];

gallery[G3, 351, 4, 9];

gallery[G3, 387, 4, 9];

gallery[G3, 423, 3, 6];

gallery[G3, 441, 2, 5];

G4

G4 = {}; lbl = {}; pr = False; 
    Do[col = Blue; 
      Do[Do[lbl = graph4Data[[ii, jj, 1]]; 
          d = graph4Data[[ii, jj, 3]]; If[pr, Print[{lbl, d}]]; 
          If[Head[d[[k]]] === String, 
            AppendTo[
              G4, {ToExpression[d[[k]]] /. 
                  regGraphOptions -> {PlotLabel -> lbl, 
                      EdgeColor -> col}}], 
            AppendTo[
              G4, {gS[4, {{1, ii}}, d[[k]]] /. 
                  regGraphOptions -> {PlotLabel -> lbl, 
                      EdgeColor -> col}}]]; grn++; 
          col = Black, {k, Length[graph4Data[[ii, jj, 3]]]}]; grt++; 
        col = Green, {jj, Length[graph4Data[[ii]]]}], {ii, 5, 
        Length[graph4Data]}]; {grn, grt, Length[G4]}

gallery[G4, 1, 5, 10];

gallery[G4, 51, 5, 10];

gallery[G4, 101, 4, 12];

gallery[G4, 149, 3, 5];

gallery[G4, 164, 3, 5];

gallery[G4, 179, 2, 5];

G5

G5 = {}; pr = False; 
    Do[col = Blue; 
      Do[Do[lbl = graph5Data[[ii, jj, 1]]; 
          d = graph5Data[[ii, jj, 3]]; If[pr, Print[{lbl, d}]]; 
          If[Head[d[[k]]] === String, 
            AppendTo[
              G5, {ToExpression[d[[k]]] /. 
                  regGraphOptions -> {PlotLabel -> lbl, 
                      EdgeColor -> col}}], 
            AppendTo[
              G5, {gS[5, {{1, ii}}, d[[k]]] /. 
                  regGraphOptions -> {PlotLabel -> lbl, 
                      EdgeColor -> col}}]]; grn++; 
          col = Black, {k, Length[graph5Data[[ii, jj, 3]]]}]; grt++; 
        col = Green, {jj, Length[graph5Data[[ii]]]}], {ii, 6, 
        Length[graph5Data]}]; {grn, grt, Length[G5]}

gallery[G5, 1, 5, 6];

gallery[G5, 31, 3, 5];

G6

G6 = {}; pr = False; 
    Do[col = Blue; 
      Do[Do[lbl = graph6Data[[ii, jj, 1]]; 
          d = graph6Data[[ii, jj, 3]]; If[pr, Print[{lbl, d}]]; 
          If[Head[d[[k]]] === String, 
            AppendTo[
              G6, {ToExpression[d[[k]]] /. 
                  regGraphOptions -> {PlotLabel -> lbl, 
                      EdgeColor -> col}}], 
            AppendTo[
              G6, {gS[6, {{1, ii}}, d[[k]]] /. 
                  regGraphOptions -> {PlotLabel -> lbl, 
                      EdgeColor -> col}}]]; grn++; 
          col = Black, {k, Length[graph6Data[[ii, jj, 3]]]}]; grt++; 
        col = Green, {jj, Length[graph6Data[[ii]]]}], {ii, 6, 
        Length[graph6Data]}]; {grn, grt, Length[G6]}

gallery[G6, 1, 5, 7];

gallery[G6, 36, 3, 3];

gallery[G6, 45, 3, 3];

G7

G7 = {}; pr = False; 
    Do[col = Blue; 
      Do[Do[lbl = graph7Data[[ii, jj, 1]]; 
          d = graph7Data[[ii, jj, 3]]; If[pr, Print[{lbl, d}]]; 
          If[Head[d[[k]]] === String, 
            AppendTo[
              G7, {ToExpression[d[[k]]] /. 
                  regGraphOptions -> {PlotLabel -> lbl, 
                      EdgeColor -> col}}], 
            AppendTo[
              G7, {gS[7, {{1, ii}}, d[[k]]] /. 
                  regGraphOptions -> {PlotLabel -> lbl, 
                      EdgeColor -> col}}]]; grn++; 
          col = Black, {k, Length[graph7Data[[ii, jj, 3]]]}]; grt++; 
        col = Green, {jj, Length[graph7Data[[ii]]]}], {ii, 8, 
        Length[graph7Data]}]; {grn, grt, Length[G7]}

gallery[G7, 1, 5, 10];

gallery[G7, 51, 3, 7];

gallery[G7, 72, 2, 3];

G8

G8 = {}; pr = False; 
    Do[col = Blue; 
      Do[Do[lbl = graph8Data[[ii, jj, 1]]; 
          d = graph8Data[[ii, jj, 3]]; If[pr, Print[{lbl, d}]]; 
          If[Head[d[[k]]] === String, 
            AppendTo[
              G8, {ToExpression[d[[k]]] /. 
                  regGraphOptions -> {PlotLabel -> lbl, 
                      EdgeColor -> col}}], 
            AppendTo[
              G8, {gS[8, {{1, ii}}, d[[k]]] /. 
                  regGraphOptions -> {PlotLabel -> lbl, 
                      EdgeColor -> col}}]]; grn++; 
          col = Black, {k, Length[graph8Data[[ii, jj, 3]]]}]; grt++; 
        col = Green, {jj, Length[graph8Data[[ii]]]}], {ii, 8, 
        Length[graph8Data]}]; {grn, grt, Length[G8]}

gallery[G8, 1, 3, 3];

Projective Plane Graphs.

See also Section PP4, PP8, PP9. PPn has a parameter f  that creates Figs4a, 4b.

pr = 0; 
    ShowGraphArray[{PPn[2, {{0, 0}}, -1], PPn[2, {{ .1,  .2}}, -1.2], 
        PPn[2], PPn[2, {{ .1,  .3}}]}];

Fig 4a. Fano Plane           Fig 4b. Fano Plane       Fig 4b. Fano Plane        Fig 4c. Fano Plane showing 
   as 6 lines and a                 with lines as                with 3 rays.                lines as complete 3-graphs by
  (yellow) triangle.                  triangles                                                      displacing the central point.

ShowLabeledGraph[PPn[5]];

SetOptions[ShowGraph, VertexNumberPosition -> { .01, 0}, 
      VertexStyle -> Disk[ .05], EdgeStyle -> Thin]; pr = 0; 
    ShowLabeledGraph[P7 = PPn[7]]; Testpp

8. graphnData Validation.

These tests create, and then check the identity, of every graph embedding in graphnData, reporting any errors.

pr = False; 
    Do[
Do[na = {3, ii, jj, {graph3Data[[ii, jj, 1]]}}; 
        d = graph3Data[[ii, jj, 3]]; If[pr, Print[{ii, jj, na, d}]]; 
        Do[no = regGraphId[graphNo[3, ii, jj, k]]; 
          If[no[[1]] ≠ na, 
            Print[{"Error in no", graph3Data[[ii, jj, k]], ii, jj, 
                no}]], 
{k, 
            Length[d]}], 
{jj, 
          Length[graph3Data[[ii]]]}], 
{ii, 6, 
        Length[graph3Data], 2}];

pr = False; 
    Do[
Do[na = {4, ii, jj, {graph4Data[[ii, jj, 1]]}}; 
        sp = graph4Data[[ii, jj, 2]]; d = graph4Data[[ii, jj, 3]]; 
        If[pr, Print[{ii, jj, na, sp, d}]]; 
        Do[no = regGraphId[tst = graphNo[4, ii, jj, k]]; 
          If[no ≠ na, 
            Print[{"Error in no", graph4Data[[ii, jj, k]], ii, jj, 
                no}]], 
{k, 
            Length[d]}], 
{jj, 
          Length[graph4Data[[ii]]]}], 
{ii, 5, 
        Length[graph4Data]}];

pr = False; 
    Do[
Do[na = {5, ii, jj, {graph5Data[[ii, jj, 1]]}}; 
        sp = graph5Data[[ii, jj, 2]]; d = graph5Data[[ii, jj, 3]]; 
        If[pr, Print[{ii, jj, na, sp, d}]]; 
        Do[no = regGraphId[tst = graphNo[5, ii, jj, k]]; 
          If[no ≠ na, 
            Print[{"Error in no", graph5Data[[ii, jj, k]], ii, jj, 
                no}]], 
{k, 
            Length[d]}], 
{jj, 
          Length[graph5Data[[ii]]]}], 
{ii, 6, 
        Length[graph5Data], 2}];

pr = False; 
    Do[
Do[na = {6, ii, jj, {graph6Data[[ii, jj, 1]]}}; 
        sp = graph6Data[[ii, jj, 2]]; d = graph6Data[[ii, jj, 3]]; 
        If[pr, Print[{ii, jj, na, sp, d}]]; 
        Do[no = regGraphId[tst = graphNo[6, ii, jj, k]]; 
          If[no ≠ na, 
            Print[{"Error in no", graph6Data[[ii, jj, k]], ii, 
                jj, "id=", no}]], 
{k, 
            Length[d]}], 
{jj, 
          Length[graph6Data[[ii]]]}], 
{ii, 6, 
        Length[graph6Data]}];

pr = False; 
    Do[
Do[na = {7, ii, jj, {graph7Data[[ii, jj, 1]]}}; 
        sp = graph7Data[[ii, jj, 2]]; d = graph7Data[[ii, jj, 3]]; 
        If[pr, Print[{ii, jj, na, sp, d}]]; 
        Do[no = regGraphId[tst = graphNo[7, ii, jj, k]]; 
          If[no ≠ na, 
            Print[{"Error in no", graph7Data[[ii, jj, k]], ii, jj, 
                no}]], 
{k, 
            Length[d]}], 
{jj, 
          Length[graph7Data[[ii]]]}], 
{ii, 7, 
        Length[graph7Data], 2}];

pr = False; 
    Do[
Do[na = {8, ii, jj, {graph8Data[[ii, jj, 1]]}}; 
        sp = graph8Data[[ii, jj, 2]]; d = graph8Data[[ii, jj, 3]]; 
        If[pr, Print[{ii, jj, na, sp, d}]]; 
        Do[no = regGraphId[tst = graphNo[8, ii, jj, k]]; 
          If[no ≠ na, 
            Print[{"Error in no", graph8Data[[ii, jj, k]], ii, jj, 
                no}]], 
{k, 
            Length[d]}], 
{jj, 
          Length[graph8Data[[ii]]]}], 
{ii, 8, 
        Length[graph8Data], 2}];

No errors found, 27/11/07.
The following routine picks up an error that easily occurs on inserting new graph isomorphs:-

gr = graph3Data; 
    Do[Do[If[Length[gr[[i, j]]] ≠ 3, 
          Print[gr[[i, j]]]], {j, Length[gr[[i]]]}], {i, 
        Length[gr]}]