(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 648961, 12222]*) (*NotebookOutlinePosition[ 650062, 12256]*) (* CellTagsIndexPosition[ 650018, 12252]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Regular Graph Synthesis, Visualization & Recognition.", "Subtitle"], Cell["Roger Beresford. 2004, revised 27th. November 2007.", "Subsubtitle"], Cell["\<\ \"The theory of symmetric cubic graphs is full of strange delights.\" Norman \ Biggs.\ \>", "Text"], Cell[CellGroupData[{ Cell["1. Summary.", "Section", PageWidth->PaperWidth], Cell[TextData[{ "\tRegular graphs have the same number ", StyleBox["edge ", FontSlant->"Italic"], "of edges meeting at each vertex (edges join different vertices). In cubic \ graphs ", StyleBox["edge", FontSlant->"Italic"], " = 3. This notebook investigates the creation, identification and display \ of small regular graphs with ", StyleBox["edge ", FontSlant->"Italic"], "= 3, 4, 5, 6, 7 or 8, in the ", StyleBox["Combinatorica", FontSlant->"Italic"], " 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.\n\tOver 810 regular \ embeddings (over 630 different graphs) are supplied in ", StyleBox["graph", FontSlant->"Italic"], "n", StyleBox["Data", FontSlant->"Italic"], ", (n=", StyleBox["edge", FontSlant->"Italic"], ", 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. ", StyleBox["regGraphId[g]", FontSlant->"Italic"], " 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 \"?\".) ", StyleBox["graphNo[ edge, size, index, isomorph]", FontSlant->"Italic"], " creates the corresponding embedding for tests or display. \n\tA few \ disconnected graphs and \"peninsula graphs\" (regular non-hamiltonian graphs \ with bridge edges) are included.\n\tProcedures are supplied to test the \ correctness of the data, and to show a \"Gallery of Graphs\". \n\tAn \ experimental procedure ", StyleBox["allGraphs", FontSlant->"Italic"], " 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.\n\tA second experimental procedure ", StyleBox["subGraph", FontSlant->"Italic"], " (inspired by [5]) attempts to remove edge sets from a known graph, and \ add others to restore regularity.\n\t", StyleBox["PPn[n]", FontSlant->"Italic"], " creates the projective plane of order ", StyleBox["n", FontSlant->"Italic"], " (if ", StyleBox["n", FontSlant->"Italic"], " is prime), and represents it as a ", StyleBox["2n+2", FontSlant->"Italic"], "-", "regular graph with coloured ", StyleBox["lines", FontSlant->"Italic"], " (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.\n\ \tPlease 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." }], "Text", PageWidth->PaperWidth] }, Closed]], Cell[CellGroupData[{ Cell["2. Version, Revisions, Notes.", "Section", PageWidth->PaperWidth], Cell[CellGroupData[{ Cell["2.1 Version.", "Subsubsection", PageWidth->PaperWidth, CellMargins->{{Inherited, -54.375}, {Inherited, Inherited}}], Cell[TextData[{ "A preliminary version was dated 24/2/4. Version 1, using ", StyleBox["Mathematica", FontSlant->"Italic"], " 5 (", StyleBox["Mathematica", FontSlant->"Italic"], " 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. ", StyleBox["newgraphs", FontSlant->"Italic"], " used to fill some gaps, 18/4/4. Bridge graphs (example 16) corrected; ", StyleBox["newgraphs", FontSlant->"Italic"], " handles 4,5,6-regular graphs. Many added. 19/4/4. Characteristic \ Polynomial, regGraphId, regGraphOptions 23/4/4. ", StyleBox["allGraphs", FontSlant->"Italic"], " added 2/5/4, replacing ", StyleBox["newgraphs", FontSlant->"Italic"], ". Used to give many new graphs 4/5/4. gS addEdge correction, additional \ graphs 4/10/4.\nMany 7-regular graphs, including Hoffman-Singleton [5], \ added; ", StyleBox["subGraph", FontSlant->"Italic"], " created in the 19/11/4 revision.\n26/11/4 More HS-subgraph symmetrical \ plots found.\n27/11/2007 More graphs added, including some 8-regular graphs \ and hyper-symmetric graphs.\nProjective Planes added, with a procedure to \ create them for prime ", StyleBox["n", FontSlant->"Italic"], " and to plot them ascoloured lines (sets of ", StyleBox["n", FontSlant->"Italic"], " edges) in ", StyleBox["2n", FontSlant->"Italic"], "-regular graphs." }], "Text", PageWidth->PaperWidth] }, Open ]], Cell[CellGroupData[{ Cell["2.2 Keywords.", "Subsubsection", PageWidth->PaperWidth, CellMargins->{{Inherited, -54.375}, {Inherited, Inherited}}], Cell["\<\ \tRegular graphs, Hamiltonian cycles, LCF, stellations, generalized Petersen \ graphs, graph embeddings, peninsula graphs, Projective Planes.\ \>", "Text", PageWidth->PaperWidth, CellMargins->{{Inherited, 1}, {Inherited, Inherited}}, CellSize->{490, Inherited}] }, Closed]], Cell[CellGroupData[{ Cell["2.3 Warnings.", "Subsubsection", PageWidth->Infinity, CellMargins->{{Inherited, -54.375}, {Inherited, Inherited}}], Cell[TextData[{ "\"Global\" effects -", StyleBox[" comb1, comb2, crash, Hc, incant, mx, tf, tst, tst1, vertx, x ", FontSlant->"Italic"], "are altered by \nvarious routines. ShowLabelledGraph and ", StyleBox["vertx", FontSlant->"Italic"], " use different labellings. Only projective planes with ", StyleBox["n", FontSlant->"Italic"], " a prime can be created, though PP4, PP8, & PP9 tables are supplied." }], "Text", PageWidth->PaperWidth, CellMargins->{{Inherited, -54.375}, {Inherited, Inherited}}] }, Closed]], Cell[CellGroupData[{ Cell["2.4 Limitations.", "Subsubsection", PageWidth->PaperWidth, CellMargins->{{Inherited, -54.375}, {Inherited, Inherited}}], Cell[TextData[{ "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 ", StyleBox["incant", FontSlant->"Italic"], " list is wrong for multi-circuit graphs. PPn vertex colouring needs \ improving." }], "Text", PageWidth->PaperWidth] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["3. Acknowledgements, References.", "Section", PageWidth->PaperWidth], Cell[TextData[{ "\tThis work was inspired by Ed Pegg's description of symmetric cubic \ graphs in ", StyleBox["Mathgames", FontSlant->"Italic"], " [2], but the procedures have the flexibility to create many others.\n\tAn \ improved version of ", StyleBox["MyIsomorphicQ", FontSlant->"Italic"], " is based on Eric Weisstein's Mathworld ", StyleBox["graphs", FontSlant->"Italic"], " package [3] (which could be wrong because ", StyleBox["Eccentricity", FontSlant->"Italic"], " gives an unsorted list). [3] also provided some LCF notations. ", StyleBox["hamiltonianCycle", FontSlant->"Italic"], " is modified from Skiena [4] p197. More interesting regular ", StyleBox["Mathworld ", FontSlant->"Italic"], "graphs added 2007." }], "Text", PageWidth->PaperWidth], Cell["[1] http://mathworld.wolfram.com/LCFNotation.html", "Text", PageWidth->PaperWidth], Cell["\<\ [2] Ed Pegg Jr, www.maa.org/editorial/mathgames/mathgames_12_29_03.html \ \>", "Text", PageWidth->PaperWidth], Cell["[3] http://mathworld.wolfram.com/graphs", "Text", PageWidth->PaperWidth], Cell["\<\ [4] Steven Skiena, Implementing Discrete Mathematics, Addison Wesley, 1990.\ \>", "Text", PageWidth->PaperWidth], Cell["[5] Ed Pegg Jr, www.maa.org/editorial/mathgames/Hoffman.gif", "Text", PageWidth->PaperWidth], Cell["[6] http://www.uwyo.edu/moorhouse/pub/planes/pg24.txt etc", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["4. Usage.", "Section", PageWidth->PaperWidth], Cell[TextData[{ "allGraphs::Usage=\"allGraphs[ed_,h_, n1_:0, n0_:{{},1}, longest_:10, \ shortest_:2] attempts to create all graphs with valency=", StyleBox["ed", FontSlant->"Italic"], ", a Hamiltonian cycle of ", StyleBox["h", FontSlant->"Italic"], " vertices, an optional inner cycle of ", StyleBox["n1", FontSlant->"Italic"], " vertices, and the crosslinks specified by ", StyleBox["n0", FontSlant->"Italic"], ". The length of arcs can be restricted by setting ", StyleBox["longest", FontSlant->"Italic"], " and ", StyleBox["shortest", FontSlant->"Italic"], ", 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.\";\n\n\ addEdge::Usage=\"addEdge[j,k] is called by other procedures to insert the \ specified edge into the ", StyleBox["comb1 & vertx", FontSlant->"Italic"], " lists. It returns ", StyleBox["True", FontSlant->"Italic"], " if successful, and ", StyleBox["False", FontSlant->"Italic"], " if one of the vertices already has a full complement of edges; this halts \ the current edge creation loop.\";\n\ncolo::Usage=\"A selection of named \ colours used by PPn. Very light and very dark colours are omitted\";\n\n\ cp::Usage=\"cp[graph] calculates the characteristic polynomial for the \ specified regular graph; the (1-", StyleBox["edge", FontSlant->"Italic"], ") term is cancelled out to reduce storage requirements.\";\n\n\ 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}.\";\n \n\ EWIsomorphicQ::Usage = \"A copy of MyIsomorphicQ from Eric Weisstein's \ Mathworld Graph package, with Eccentric corrected\";\n\n\ 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.\";\n\ngenPetersen::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. \";\n\ngP::Usage=\"Abbreviation for genPetersen\";\n\ngraphNo::Usage = \ \"graphNo[ed, no of vertices, index number, isomorph number] creates the \ specified graph, using the LCF procedure if the ", StyleBox["graphNData", FontSlant->"Italic"], " is a list and power (a common form), otherwise it uses ", StyleBox["ToExpression[string Data]", FontSlant->"Italic"], ".\"; \n\ngraphSynthesis::Usage=\"graphSynthesised[ed_,c_,croslnk_] creates \ an undirected graph in ", StyleBox["Combinatorica ", FontSlant->"Italic"], "format. The graph will have (at most) ", StyleBox["ed", FontSlant->"Italic"], " 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.\nThe ", StyleBox["croslnk", FontSlant->"Italic"], " 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.\nThe graph is \ built up as three lists:- ", StyleBox["vertx", FontSlant->"Italic"], " is the indices of the vertices linked to vertex[i] (used by ", StyleBox["hamiltonianCycle & replot", FontSlant->"Italic"], "); ", StyleBox["comb1 & comb2", FontSlant->"Italic"], " are lists of edges and points that are finally combined to give the ", StyleBox["Combinatorica", FontSlant->"Italic"], " 'Graph' structure.\";\n\ngS::Usage=\"Abbreviation for graphSynthesis\";\n\ \ngShort::Usage=\"converts a ", StyleBox["PPn", FontSlant->"Italic"], " graph to an un-coloured form that can be tested by ", StyleBox["regGraphId", FontSlant->"Italic"], " etc.", "\";\n\nhamiltonianCycle::Usage=\"hamiltonianCycle[n] uses the data in ", StyleBox["vertx", FontSlant->"Italic"], " 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. \";\n\ninfill::Usage=\"", StyleBox["allGraphs", FontSlant->"Italic"], " uses ", StyleBox["infill", FontSlant->"Italic"], " to recursively add chords\";\n\nplTopp::Usage=\"Converts a line list to a \ ", StyleBox["pp", FontSlant->"Italic"], " array\";\n\nPPn::Usage=\"PPn[", StyleBox["order,centre", FontSlant->"Italic"], "_:{{0,0}}] creates (if ", StyleBox["order", FontSlant->"Italic"], " is prime) the correspondingProjective Plane as an adjacency matrix pp, a \ line list pl, and a ", StyleBox["2n", FontSlant->"Italic"], "-", "regular graph. The graph consists of a point at ", StyleBox["centre", FontSlant->"Italic"], " and rays of points joined by edges, coloured to show line membership. \ Each possible edge appears in exactly one line. If ", StyleBox["pr", FontSlant->"Italic"], " is negative, a table ", StyleBox["ppTest", FontSlant->"Italic"], " (examples ", StyleBox["pp4, pp8, pp9", FontSlant->"Italic"], " supplied) is imported after the graph framework has been created\";\n\n\ regGraphId::Usage = \"regGraphId[graph,c_:False]; Combinatorica graphs need \ c=True to create the specified regular graph and generate ", StyleBox["vertx", FontSlant->"Italic"], ". ", StyleBox["vertx", FontSlant->"Italic"], " 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.\";\n\nreplot::Usage=\"replot[n,a] creates a graph from the ", StyleBox["n", FontSlant->"Italic"], "'th circuit in Hc, as produced by ", StyleBox["hamiltonianCycle", FontSlant->"Italic"], ". 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 ", StyleBox["a", FontSlant->"Italic"], " to improve the appearance.\";" }], "Input", PageWidth->PaperWidth, Evaluatable->False], Cell[BoxData[ RowBox[{ RowBox[{\(subGraph::Usage\), "=", "\"\\"Italic\"]\) to include neighbours to \ the specified depth, and selects (positive depth) or deletes (non-positive \ depth) them before adding the \!\(\* StyleBox[\"inlist\",\nFontSlant->\"Italic\"]\) edges. The graph is stored as \ \!\(\* StyleBox[\"tst1\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\"and\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\"is\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\"plotted\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\"unless\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\"depth\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\"is\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Plain\"]\)real\!\(\* StyleBox[\".\",\nFontSlant->\"Plain\"]\)\>\""}], ";", RowBox[{\(testGraph::Usage\), "=", "\"\<\!\(\* StyleBox[\"testGraph\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\"is\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\"used\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\"by\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\"allGraphs\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"&\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"infill\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\"to\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\"test\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\"for\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\"unknown\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\"graphs\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\"and\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\"to\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\"store\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\"new\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\"ones\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\"in\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\"results\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"&\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"incant\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\".\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\"incant\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\"is\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\"still\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\"wrong\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\"for\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\"multi\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\"-\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\"circuits\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\".\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\"\\\"\",\nFontSlant->\"Plain\"]\)"}], ";", RowBox[{\(Testpp::Usage\), "=", "\"\\"Italic\"]\).\>\""}], ";"}]], "Input", PageWidth->PaperWidth] }, Closed]], Cell[CellGroupData[{ Cell["\<\ 5. Programmes and Data, Version dated 27/11/7. 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As it needs \ ", StyleBox["CharacteristicPolynomial", FontSlant->"Italic"], " from ", StyleBox["Mathematica", FontSlant->"Italic"], "5 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.\n\tFor regular graphs \ use the faster \"", StyleBox["regIsomorphicQ", FontSlant->"Italic"], "\"", StyleBox[", ", FontSlant->"Italic"], "which omits irrelevant tests." }], "Text", PageWidth->PaperWidth], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ StyleBox["eccSort", FontColor->RGBColor[1, 0, 0]], "[", "a_", "]"}], ":=", \(Sort[Eccentricity[a]]\)}], ";", \( (*\(16/4\)/4*) \)}]], "Input", PageWidth->PaperWidth], Cell[TextData[{ StyleBox["EWIsomorphicQ", FontColor->RGBColor[1, 0, 0]], "[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]" }], "Input", PageWidth->PaperWidth], Cell[TextData[{ StyleBox["regIsomorphicQ", FontColor->RGBColor[1, 0, 0]], "[g_Graph, h_Graph] := Module[{x}, \ CharacteristicPolynomial[ToAdjacencyMatrix[g], x] === \ CharacteristicPolynomial[ToAdjacencyMatrix[h], x]]" }], "Input", PageWidth->PaperWidth], Cell[TextData[{ StyleBox["toVertx", FontColor->RGBColor[1, 0, 0]], "[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];" }], "Input", PageWidth->PaperWidth], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ StyleBox["gallery", FontColor->RGBColor[1, 0, 0]], "[", \(G_: G6, strt_: 1, ni_: 4, nj_: 8\), "]"}], ":=", \(ShowGraphArray[ Table[G[\([strt - 1 + i + ni \((j - 1)\), 1]\)], {j, nj}, {i, ni}]]\)}], ";"}]], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(\(regGraphOptions = {};\)\)], "Input"], Cell[BoxData[ \(\(\(SetOptions[ShowGraph, VertexColor \[Rule] Red, VertexStyle \[Rule] Disk[ .06], VertexNumberColor \[Rule] Black, VertexNumberPosition \[Rule] { .02, 0}, \ EdgeColor \[Rule] Black, PlotRange \[Rule] All];\)\( (*\ Modified\ from\ EW' s\ Graphs\ package\ to\ put\ numbers\ inside\ vertices*) \)\)\)], \ "Input", PageWidth->PaperWidth] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["Graph Data.", "Subsection", PageWidth->PaperWidth], Cell["\<\ 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.\ \>", "Text", PageWidth->PaperWidth], Cell[CellGroupData[{ Cell["3 regular", "Subsubsection", PageWidth->PaperWidth], Cell["\<\ 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. \ \>", "Text"], Cell[BoxData[ RowBox[{\(x =. \), ";", RowBox[{"Clear", "[", StyleBox["graph3Data", FontColor->RGBColor[1, 0, 0]], "]"}], ";", RowBox[{ StyleBox["graph3Data", FontColor->RGBColor[1, 0, 0]], "=", RowBox[{"{", RowBox[{\({}\), ",", \({}\), ",", \({}\), ",", RowBox[{"{", StyleBox[\( (*4*) \), FontColor->RGBColor[0, 1, 0]], \({"\<3,4,24\>", \((1 + x)\)\^3, {{{2, 2}, 2}}}\), "}"}], ",", \({}\), ",", "\[IndentingNewLine]", RowBox[{"{", StyleBox[\( (*6*) \), FontColor->RGBColor[0, 1, 0]], \({"\<3,6,72\>", x\^4\ \((3 + x)\), {{{3}, 3}, "\"}}, {"\<3,6,12\>", \((\(-1\) + x)\)\ x\^2\ \((2 + x)\)\^2, {{{\(-2\), 2, 3}, 1, 1}, "\", "\", \ "\"}}\), "}"}], ",", \({}\), ",", "\[IndentingNewLine]", RowBox[{"{", StyleBox[\( (*8*) \), FontColor->RGBColor[0, 1, 0]], \({"\<3,8,48\>", \((\(-1\) + x)\)\^3\ \((1 + x)\)\^3\ \ \((3 + x)\), {{{\(-3\), 3}, 2}, "\", \ "\"}}, \ \[IndentingNewLine]{"\<3,8,16,1\>", \((\(-1\) + x)\)\^2\ \((1 + x)\)\ \((\(-1\) + 2\ x + x\^2)\)\^2, {{{4}, 4}, {{\(-3\), 3, 4, 4}, 1}, "\"}}, \[IndentingNewLine]{"\<3,8,16,2\>", \ \((\(-1\) + x)\)\ \((1 + x)\)\^4\ \((\(-5\) + x\^2)\), {{{2, 2}, 3}, "\"}}, \[IndentingNewLine]{"\<3,8,12\>", x\ \((\(-4\) + x + x\^2)\)\ \((\(-1\) + x + x\^2)\)\^2, {{{\(-3\), \(-3\), 2, 4}, 1, 1}}}, \[IndentingNewLine]{"\<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}, "\", "\"}}, \ \[IndentingNewLine]{"\<3,8,nx\>", \((\(-3\) + x)\)\ \((1 + x)\)\^6, \ {"\"}}\), "}"}], ",", \({}\), ",", "\[IndentingNewLine]", RowBox[{"{", StyleBox[\( (*10*) \), FontColor->RGBColor[0, 1, 0]], \({"\", \((\(-1\) + x)\)\^5\ \((2 + \ x)\)\^4, {"\", \ "\", \ "\", \ "\", \ "\"}}, \ \[IndentingNewLine]{"\<3,10,48\>", \((\(-2\) + x)\)\ \((\(-1\) + x)\)\^2\ x\^2\ \((1 + x)\)\^2\ \((2 + x)\)\ \((3 + x)\), {{{5, \(-3\), \(-3\), 3, 3}, 1}}}, \[IndentingNewLine]{"\<3,10,20,1\>", \((3 + x)\)\ \((\(-1\) - 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3\ x + x\^3)\)\ \^2, {{{6, \(-4\), 5, \(-4\), 4, 5}, 1}}}, \[IndentingNewLine]{"\<3,12,16,1\>", \((\(-2\) + x)\)\ \((\(-1\) + x)\)\^2\ x\^3\ \((2 + x)\)\^3\ \ \((\(-4\) + x + x\^2)\), {{{\(-5\), 5, 6}, 2}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<3,12,16,4\>", \((\(-2\) + x)\)\ \((\(-1\) + x)\)\ x\^3\ \((1 + x)\)\ \((2 + x)\)\^3\ \((\(-4\) - x + x\^2)\), {{{\(-2\), 2, \(-3\)}, 3}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<3,12,12,1\>", \((\(-1\) + x)\)\ x\^2\ \((2 + x)\)\^2\ \((\(-5\) + x\^2)\)\ \((\(-2\) + x\^2)\)\^2, {"\", \ "\", "\", {{\(-4\), \ \(-2\), 5, \(-5\), 2, 4}, 1}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<3,12,12,nh\>", \ \((\(-2\) + x)\)\ \((\(-1\) + x)\)\^2\ \((1 + x)\)\^3\ \((2 + x)\)\ \((\(-3\) + x + x\^2)\)\^2, \ {"\", \ "\", \ "\"}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<3,12,8,4\>", \((\(-1\) + x)\)\ \((\(-3\) + x\^2)\)\ \((\(-1\) + 2\ x + x\^2)\)\ \((\(-1\) - 3\ x + x\^2 + x\^3)\)\^2, \ {"\"}}, \ \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, {"\", \((\(-2\) + 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}, "\"}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}, \[IndentingNewLine]{{\(-4\), \(-3\), 3, \(-4\), 3, 0, 3}, 1}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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)\), {"\"}}, \ \[IndentingNewLine]{"\<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)\), \ {"\"}}, \ \[IndentingNewLine]{"\<3,12,4,8\>", \((\(-2\) + x)\)\^2\ x\ \((1 + x)\)\^3\ \ \((2 + x)\)\ \((\(-2\) + x\^2)\)\ \((\(-2\) + 2\ x + x\^2)\), \ {"\"}}, \ \[IndentingNewLine]{"\<3,12,4,9\>", \((\(-2\) + x)\)\ \((\(-1\) + x)\)\ \((1 + x)\)\^3\ \((\(-3\) + x + x\^2)\)\ \((2 - 5\ x - 4\ x\^2 + 2\ x\^3 + x\^4)\), \ {"\"}}, \ \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}, "\"}}, \ \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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\ "\"\ \ graphs*) }, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}, "\"}}, \[IndentingNewLine]{"\<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}, \[IndentingNewLine]{{\(-4\), 6, \(-3\), 6, 2, 5}, 1}, {{\(-5\), \(-5\), \(-3\), 6, 2, 5}, 1}}}, \[IndentingNewLine]{"\", \((\(-2\) + 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}, "\"}}, \[IndentingNewLine]{"\", x\^3\ \((1 + x)\)\ \((2 + x)\)\ \((\(-2\) + x\^2)\)\ \((14 - 2\ x - 9\ x\^2 + x\^4)\), \ {"\"}},\ \[IndentingNewLine]{"\", x\^4\ \((1 + x)\)\ \((\(-2\) + 2\ x + x\^2)\)\ \((14 - 2\ x - 9\ x\^2 + x\^4)\), \ {"\"}}, \ \[IndentingNewLine]{"\", x\ \((1 + x)\)\^3\ \((\(-1\) + x + x\^2)\)\ \((\(-12\) + 23\ x + 7\ x\^2 - 10\ x\^3 - x\^4 + x\^5)\), \ {"\"}}, \[IndentingNewLine]{"\", x\^2\ \((1 + x)\)\^2\ \((\(-1\) + x + x\^2)\)\ \((\(-13\) + 26\ x + x\^2 - 11\ x\^3 + x\^5)\), \ {"\"}}\), "}"}], ",", \({}\), ",", "\[IndentingNewLine]", RowBox[{"{", StyleBox[\( (*14*) \), FontColor->RGBColor[0, 1, 0]], \({"\", \((3 + x)\)\ \((\(-2\) + x\^2)\)\^6, {{{\(-5\), 5}, 7, \(-2\)}, \ "\"}}, \ \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine] (*\({"\<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}}}\)\(,\)*) \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}, "\", "\", \ "\"}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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, \(-1\)}}}, \[IndentingNewLine]{"\<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, \(-1\)}}}, \[IndentingNewLine]{"\<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, \(-1\)}}}, \[IndentingNewLine]{"\<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, \(-4\)}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<3,14,8,7\>", \((\(-2\) + \ x\^2)\)\^4\ \((\(-2\) + 2\ x + x\^2)\)\ \((\(-2\) - 4\ x + x\^2 + x\^3)\), \ {"\"}}, \ \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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, \(-1\)}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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, \(-3\)}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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)\), \ {"\"}}, \ \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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, \(-3\)}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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, \(-2\)}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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)\), \ {"\"}}, \ \[IndentingNewLine]{"\<3,14,2,11\>", \((\(-1\) + 3\ x + 11\ x\^2 - 8\ x\^3 - 6\ x\^4 + 2\ x\^5 + x\^6)\)\ \((1 - 12\ x + 8\ x\^2 + 19\ x\^3 - 6\ x\^4 - 8\ x\^5 + x\^6 + x\^7)\), \ {"\"}}, \ \[IndentingNewLine]{"\<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 - 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8\ x\^6 - 43\ x\^7 - 7\ x\^8 + 4\ x\^9 + x\^10)\), {{{\(-3\), \(-2\), 6, 6, 6, 2, 6}, 1}}}, \[IndentingNewLine]{"\<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, \(-2\)}}}, \[IndentingNewLine]{"\<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, \(-2\)}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<3,14,1,7\>", \((\(-2\) + \ x\^2)\)\^2\ \((3 + 5\ x - 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x\^2)\)\ \((\(-1\) + x + x\^2)\)\^2\ \((1 + 3\ x + x\^2)\), {{{2, 3}, 9}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}, "\"}}, \ \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}\), "}"}], ",", "\[IndentingNewLine]", RowBox[{"{", StyleBox[\( (*10*) \), FontColor->RGBColor[0, 1, 0]], \({"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\", x\ \((1 + x)\)\^4\ \((\(-5\) + x\^2)\)\^2, \ {"\", {{\(-2\), 2}, 5}, {{2, 3, \(-3\), \(-2\)}, 5}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}, "\", "\"}}, \ \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<4,10,10\>", x\ \((5 - 5\ x - 4\ x\^2 + 2\ x\^3 + x\^4)\)\^2, {{{4, 2}, 5}, "\", {{8, 5, 3, 0, 3, 0, 0, 0, 0, 0, 6, 2, 6, 4, 5, 4, 4, 0, 2}, 1}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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, \(-2\)}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<4,10,4,4\>", \((\(-1\) + \ x)\)\^2\ x\ \((1 + x)\)\^3\ \((3 + x)\)\ \((\(-5\) + x\^2)\), {{{2, \(-3\), \(-3\), \(-2\), \(-4\)}, 2, 3}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<4,10,4,7?\>", (*\ isospecral\ non - isomorphicp6*) \[IndentingNewLine]\((\(-1\) + 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}, \[IndentingNewLine]"\", \ "\", \ "\"}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}, "\"}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<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}}}, \[IndentingNewLine]{"\<4,10,1,2\>", \((\(-1\) \ + x)\)\ \((1 + x)\)\^2\ \((\(-4\) + 23\ x + 9\ x\^2 - 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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, \ {"\"}}\), "}"}]}], "}"}]}], ";"}]], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["8 regular", "Subsubsection", PageWidth->PaperWidth], Cell[BoxData[ RowBox[{\(x =. \), ";", RowBox[{ StyleBox["graph8Data", FontColor->RGBColor[1, 0, 0]], "=", RowBox[{"{", RowBox[{\({}\), ",", \({}\), ",", \({}\), ",", \({}\), ",", \({}\), ",", \({}\), ",", \({}\), ",", "\[IndentingNewLine]", \({}\), ",", RowBox[{"{", StyleBox[\( (*9*) \), FontColor->RGBColor[0, 1, 0]], RowBox[{\({"\<8,9 362880\>", \(-\((1 + x)\)\^8\), {{{2, 3, 4, \(-3\), \(-2\)}, 9}}}\), Cell[""]}], "}"}], ",", \({}\), ",", RowBox[{"{", StyleBox[\( (*11*) \), FontColor->RGBColor[0, 1, 0]], \({"\<8,11,22\>", \(-\((1 - 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It creates a cycle of ", StyleBox["v", FontSlant->"Italic"], " vertices, with an optional inner cycle having symmetrical links to the \ outer circuit.The chords specified in ", StyleBox["n0", FontSlant->"Italic"], " (using any of the ", StyleBox["croslnk", FontSlant->"Italic"], " formats) are", " then added. If this does not give a regular graph, ", StyleBox["infill", FontSlant->"Italic"], " is called with arguments ", StyleBox["vertx", FontSlant->"Italic"], " and ", StyleBox["done,", FontSlant->"Italic"], " an (initially empty) list of added edges. A list ", StyleBox["new", FontSlant->"Italic"], " of acceptable chords is created. These are sequentially added to the \ partly filled ", StyleBox["vertx", FontSlant->"Italic"], ", and ", StyleBox["infill", FontSlant->"Italic"], " 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 ", StyleBox["pr", FontSlant->"Italic"], " can be set to 1, whereupon all regular graphs are stored.\n\tThe 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." }], "Text", PageWidth->PaperWidth], Cell[TextData[{ "Colour coding:-\n", StyleBox["green", FontColor->RGBColor[0, 1, 0]], " - module names, calls, and returns, ", StyleBox["red", FontColor->RGBColor[1, 0, 0]], " - error failures, ", StyleBox["blue", FontColor->RGBColor[0, 0, 1]], " - diagnostic print messages. ", StyleBox["pr", FontSlant->"Italic"], " is a global variable that controls the amount of ", StyleBox["diagnostic output", FontColor->RGBColor[0, 0, 1]], ", with graphs being shown as they are developed if ", StyleBox["pr", FontSlant->"Italic"], ">2. (It is True or False when creating the embeddings in Section 7; all \ output is suppressed in the False case.) If ", StyleBox["pr", FontSlant->"Italic"], " is 1, all graphs go into ", StyleBox["results", FontSlant->"Italic"], "; otherwise ", StyleBox["results", FontSlant->"Italic"], " only contains graphs that are NOT in the database.\n\nStrategy:- \n", StyleBox["allGraphs", FontColor->RGBColor[0, 1, 0]], ". Set up global variables - ", StyleBox["v,nv ", FontSlant->"Italic"], "= no. of vertices, ", StyleBox["results, incant ", FontSlant->"Italic"], "(incantations that will create each new graph),", StyleBox[" mx", FontSlant->"Italic"], " (the maximum number of edge ends at each vertex)", StyleBox[", depth", FontSlant->"Italic"], " (a diagnostic indication of the recursion depth), and", StyleBox[" kapol ", FontSlant->"Italic"], "(a list of known characteristic polynomials).\nCreate the prescribed ", StyleBox["tst", FontSlant->"Italic"], " 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 ", StyleBox["vertx", FontSlant->"Italic"], ", the lists of vertices connected to each vertex.", StyleBox["\n", FontSlant->"Italic"], "If ", StyleBox["vertx", FontSlant->"Italic"], " is complete, call ", StyleBox["testGraph", FontColor->RGBColor[0, 1, 0]], "; otherwise call ", StyleBox["infill", FontColor->RGBColor[0, 1, 0]], " to find the sets of chords that create complete graphs, as follows:-\nDo \ 1\[LessEqual]i\[LessEqual]nv-short;\n If vertx[[i]] incomplete, Do i+short\ \[LessEqual]j\[LessEqual]nv \n If vertx[[j]] incomplete & chord not too \ long, add the edge and skip out of the Do loops. \n Test any complete graph, \ backtrack if the Do loops have not found a new edge, or call ", StyleBox["infill", FontColor->RGBColor[0, 1, 0]], " recursively. ", StyleBox["vertx, done & comb1", FontSlant->"Italic"], " have to be restored in this case.\nComplete graphs are tested in ", StyleBox["testGraph", FontColor->RGBColor[0, 1, 0]], "; unknown graphs are stored in ", StyleBox["results", FontSlant->"Italic"], " and their incantations in ", StyleBox["incant", FontSlant->"Italic"], "; their characteristic polynomials will be added to ", StyleBox["kapol", FontSlant->"Italic"], ". If ", StyleBox["pr", FontSlant->"Italic"], "=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.\nOn returning to ", StyleBox["allGraphs", FontColor->RGBColor[0, 1, 0]], ", the number of new graphs is reported. They can be viewed by ", StyleBox["ShowGraph[results[[n]]]; ", FontSlant->"Italic"], "or", " ", StyleBox["ShowGraphArray[Partition[results,?]];", FontSlant->"Italic"], " and added (manually) to ", StyleBox["graphNData.", FontSlant->"Italic"] }], "Text", PageWidth->PaperWidth] }, Open ]], Cell[CellGroupData[{ Cell["allGraphs.", "Subsubsection", PageWidth->PaperWidth], Cell[BoxData[ RowBox[{\( (*This\ attempts\ to\ create\ all\ unknown\ ed, h, \((optional\ inner, optional\ chord - length\ range)\) graphs\ with\ some\ specified\ edges*) \), "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ StyleBox["allGraphs", FontColor->RGBColor[0, 1, 0]], "[", \(ed_, h_, n1_: 0, n0_: {{0}, 1}, longest_: 10, shortest_: 2\), "]"}], ":=", RowBox[{"Module", "[", RowBox[{\({r = 0, v, x = 0, inner = Abs[n1]}\), ",", RowBox[{ StyleBox[\(If[\(! 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In many \ cases the sub-graph is regular with a lower number of edges per vertex. In \ others, some neighbours of deleted vertices may need to be joined to maintain \ regularity. This is done by inspection; programming a symmetrical join could \ be difficult.\n\tStrategy - create a known labelled graph as ", StyleBox["tst", FontSlant->"Italic"], ". Call ", StyleBox["subGraph", FontSlant->"Italic"], " with a list of vertices and a depth - the depth to which their neighbours \ are to be found. If ", StyleBox["depth", FontSlant->"Italic"], " is negative, these are to be deleted. If ", StyleBox["depth", FontSlant->"Italic"], " is positive, they are to be selected, deleting all others. The final \ deletion list is used to give ", StyleBox["nc1, nc2", FontSlant->"Italic"], " for the revised graph by selecting edges from ", StyleBox["comb1", FontSlant->"Italic"], " and vertices from ", StyleBox["comb2", FontSlant->"Italic"], ". The edges in ", StyleBox["inlist", FontSlant->"Italic"], " (using the vertex numbers of the original graph) are then added. The \ number of edges per vertex is calculated, and the result is stored as ", StyleBox["tst1", FontSlant->"Italic"], " and plotted unles ", StyleBox["depth", FontSlant->"Italic"], " is real. 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IntegerQ[mx]\), Print[{"\", Length[tst1[\([1]\)]], "\", Length[tst1[\([2]\)]]}]]\), ";", "\[IndentingNewLine]", \(If[IntegerQ[depth], ShowLabeledGraph[tst1]]\), ";", "tst1"}]}], "]"}]}], ";"}]], "Input", PageWidth->PaperWidth], Cell[TextData[{ StyleBox["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. 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Point \ colouring needs more work." }], "Text"], Cell[BoxData[{ \(\(SetOptions[ShowGraph, VertexStyle \[Rule] Disk[ .06], VertexNumberPosition \[Rule] { .02, 0}, \ EdgeColor \[Rule] Black, PlotRange \[Rule] All, EdgeStyle \[Rule] Normal];\)\), "\n", \(\(colo = ToExpression[{"\", "\", "\", "\", \ "\", "\", "\", "\", "\", \ "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\ \", "\", "\", "\", \ "\", "\", "\", "\", \ "\", "\", "\", "\", \ "\", "\", "\", "\", "\ \", "\", "\", "\", \ "\", "\", "\", "\", \ "\", "\", "\", "\", \ "\", "\", "\", \ "\", "\", "\", \ "\", "\", "\", "\", \ "\", "\", "\", "\", \ "\", "\", "\", "\", \ "\", "\", "\", "\", \ "\", "\", "\", "\", \ "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", "\", \ "\", "\", "\", \ "\", "\", "\", "\", \ "\", "\", "\", "\", "\", \ "\", "\", "\", "\", \ "\", "\", "\", "\", \ "\", "\", "\", \ "\", "\", "\", "\", \ "\", "\", "\", "\", \ "\", "\", "\", "\", \ "\", "\", "\", "\", \ "\", "\", "\", "\", "\", \ "\", "\", "\", "\", \ "\", "\", "\", "\", \ "\", "\", "\", \ "\", "\", "\", "\", \ "\", "\"}];\)\)}], "Input"], Cell[BoxData[ FormBox[ RowBox[{\(Clear[ProjectivePlaneN, PPn, PPtest, pr]\), ";", \(ProjectivePlaneN[any___] := PPn[any]\), ";", RowBox[{\(PPn[n_, cntr_: {{0, 0}}, f_: 1]\), ":=", RowBox[{"(*", RowBox[{ RowBox[{ RowBox[{"Create", " ", StyleBox[ RowBox[{"PP", StyleBox["n", FontSlant->"Italic"]}]], " ", "Table"}], " ", "&"}], " ", "Graph"}], "*)"}], "\[IndentingNewLine]", RowBox[{"Module", "[", RowBox[{\({angl = If[EvenQ[n], 0, \[Pi]/\((n + 1)\), 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. \[Pi]/\((n + 1)\)}\), ",", "\[IndentingNewLine]", RowBox[{\(c = Range[m]\), ";", \(pp = Table[0, {i, m}, {j, m}]\), \( (*Global*) \), ";", \(pl = {}\), \( (*Global*) \), ";", "\[IndentingNewLine]", StyleBox[\( (*Create\ points\ on\ rays*) \), FontColor->RGBColor[0, 0, 1]], "\[IndentingNewLine]", \(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}]\), ";", "\[IndentingNewLine]", StyleBox[\( (*Create\ line\ Array*) \), FontColor->RGBColor[0, 0, 1]], \(pp[\([1, 1]\)] = 1\), ";", "\[IndentingNewLine]", \(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}]\), ";", StyleBox[\( (*First\ n1\ rows\ and\ columns\ are\ now\ \ complete\ *) \), FontColor->RGBColor[0, 0, 1]], "\[IndentingNewLine]", \(ix = 0\), ";", RowBox[{"Do", "[", RowBox[{ RowBox[{\(i0 = n\ i + 1\), ";", \(iy = 0\), ";", StyleBox["\[IndentingNewLine]", FontColor->RGBColor[1, 0, 0]], RowBox[{"Do", "[", RowBox[{ RowBox[{\(j0 = n\ j + 1\), " ", ";", " ", StyleBox[\( (*Key\ \(Calculation!\)\ *) \), FontColor->RGBColor[0, 0, 1]], StyleBox["\[IndentingNewLine]", FontColor->RGBColor[1, 0, 1]], RowBox[{"Do", "[", RowBox[{ RowBox[{ RowBox[{"v", "=", StyleBox[\(j0 + Mod[l + ix\ iy, n, 1]\), FontColor->RGBColor[1, 0, 0]]}], StyleBox[";", FontColor->RGBColor[1, 0, 0]], StyleBox[\(pp[\([i0 + Mod[l, n, 1], v]\)] = v\), FontColor->RGBColor[1, 0, 0]]}], ",", \({l, n}\)}], "]"}], ";", "\[IndentingNewLine]", \(iy++\)}], ",", \({j, n}\)}], "]"}], ";", StyleBox["\[IndentingNewLine]", FontColor->RGBColor[1, 0, 1]], \(ix++\)}], ",", \({i, n}\)}], "]"}], StyleBox[";", FontColor->GrayLevel[0]], StyleBox["\[IndentingNewLine]", FontColor->GrayLevel[0]], StyleBox[\( (*pr > 1\ outputs\ pp; \ negative\ pr\ inputs\ a\ test\ array*) \), FontColor->RGBColor[0, 0, 1]], "\[IndentingNewLine]", \(If[pr > 1, Print[pp // tf]]\), ";", \(If[pr < 0, pp = ppTest; Print["\"];]\), ";", "\[IndentingNewLine]", StyleBox[\( (*Create\ a\ compact\ set\ "\" of\ closed\ lines\ from\ "\"*) \), FontColor->RGBColor[0, 0, 1]], StyleBox["\[IndentingNewLine]", FontColor->RGBColor[1, 0, 0]], RowBox[{"Do", "[", RowBox[{ RowBox[{\(p = Rest[Union[pp[\([i]\)]]]\), ";", \(AppendTo[pl, p]\), ";", \(AppendTo[p, First[p]]\), ";", StyleBox["\[IndentingNewLine]", FontColor->RGBColor[1, 0, 0]], \(k = 1\), ";", RowBox[{ "While", "[", \(k <= Length[p] && \(! 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Tutorial.", "Section", PageWidth->PaperWidth], Cell[CellGroupData[{ Cell["Pre-defined Graphs, Combinatorica Instructions.", "Subsection"], Cell[TextData[{ "\tThe ", StyleBox["graph", FontSlant->"Italic"], "K", StyleBox["Data", FontSlant->"Italic"], " files contain incantations for over 820 graph embeddings, which can be \ shown via Combinatorica instructions and ", StyleBox["graphNo[K:3, v:6, counter:1, incantation:1];", FontSlant->"Italic"], " ", StyleBox["K", FontSlant->"Italic"], " is the number of edges per vertex (3 to 7), ", StyleBox["v", FontSlant->"Italic"], " is the number of vertices, ", StyleBox["counter", FontSlant->"Italic"], " is the position in the specified {K,v} list (which is ordered by \ decreasing symmetry), and ", StyleBox["incantation", FontSlant->"Italic"], " selects from several embeddings (if present)." }], "Text", PageWidth->PaperWidth], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*Example\ 0\ *) \)\(Print[{"\", \[IndentingNewLine]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] /. \[IndentingNewLine]{regGraphOptions \[Rule] {EdgeColor \ \[Rule] Blue, VertexNumberColor \[Rule] Red, VertexNumberPosition \[Rule] { .02, 0}, \ VertexColor \[Rule] Black, VertexStyle \[Rule] Disk[ .07], PlotLabel -> "\<3,6,2,3\>"}}];\[IndentingNewLine] \(ShowGraphArray[{tst1, tst2, tst3, tst4}];\)\)\)\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ \({"Data for the two 3-regular 6-vertex graphs:-\n", {{"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}}]"}}}}\)], "Print"] }, Open ]], Cell["Session changes can be made to graph appearences by e.g.", "Text", PageWidth->PaperWidth], Cell[BoxData[ \(\(regGraphOptions = {EdgeColor \[Rule] Blue, VertexNumberColor \[Rule] Red, VertexNumberPosition \[Rule] { .02, 0}, \ VertexColor \[Rule] Black, VertexStyle \[Rule] Disk[ .07]};\)\)], "Input", PageWidth->PaperWidth] }, Closed]], Cell[CellGroupData[{ Cell["Graph creation.", "Subsection"], Cell[TextData[{ "\t", StyleBox["graphSynthesis[]", FontSlant->"Italic"], " uses the default parameters ", StyleBox["ed_:3, c_:{{.5,5,2},{1,5}}, croslnk_:{}", FontSlant->"Italic"], " 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:-" }], "Text", PageWidth->PaperWidth], Cell[BoxData[ \(\(\( (*Example\ 1*) \)\(ShowLabeledGraph[ graphSynthesis[]];\)\)\)], "Input", PageWidth->PaperWidth], Cell[TextData[{ "\tGraph properties can be changed by changing the ", StyleBox["regGraphOptions:-", FontSlant->"Italic"] }], "Text"], Cell[BoxData[ \(\(\( (*Example\ 1 a*) \)\(ShowLabeledGraph[ graphSynthesis[] /. {regGraphOptions \[Rule] {EdgeColor \[Rule] Blue, VertexNumberColor \[Rule] White, VertexNumberPosition \[Rule] { .02, 0}, \ VertexColor \[Rule] Black, VertexStyle \[Rule] Disk[ .07]}}];\)\)\)], "Input", PageWidth->PaperWidth], Cell[TextData[{ "\tThe number of edges per vertex is given by ", StyleBox["ed_:3.", FontSlant->"Italic"], " The inner circuit is defined by ", StyleBox["{.5,5,2}", FontSlant->"Italic"], " 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.\n\tNow make \ some changes. The default options apply automatically. Changing ", StyleBox["ed", FontSlant->"Italic"], " 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 -\[Pi]/5 to give a pretty picture." }], "Text", PageWidth->PaperWidth], Cell[BoxData[ \(\(\( (*Example\ 2*) \)\(\(newgraph = graphSynthesis[ 4, {{1, 5}, { .5, 5, 3, \(-\[Pi]\)/5. }}];\)\[IndentingNewLine] \(ShowLabeledGraph[newgraph];\)\)\)\)], "Input", PageWidth->PaperWidth], Cell["\<\ \tNext, 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.)\ \>", "Text", PageWidth->PaperWidth], Cell[BoxData[ \(\(\( (*Example\ 3*) \)\(pr = 3; ShowLabeledGraph[ graphSynthesis[3, {{1, 9}}, {{1, 3, \(-1\)}}]];\)\)\)], "Input", PageWidth->PaperWidth], Cell["\<\ \tChange 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:-\ \>", "Text", PageWidth->PaperWidth], Cell[BoxData[ \(\(\( (*Example\ 3 a*) \)\(nonagon4 = graphSynthesis[4, {{1. , 9}}, {{1, 3, \(-1\)}}]; ShowLabeledGraph[nonagon4]; regGraphId[nonagon4]\n Automorphisms[nonagon4]\)\)\)], "Input", PageWidth->PaperWidth], Cell[TextData[{ "The data entry shows that it is the first entry in the ", StyleBox["graph4Data", FontSlant->"Italic"], " list, having 18 automorphisms and a ", StyleBox["CharacteristicPolynomial", FontSlant->"Italic"], " with 5 factors (the (x-4) factor of a 4-regular graph is implicit). The \ incantation is an LCF4 notation {", Cell[BoxData[ \(TraditionalForm\`3\^9\)]], "} :-" }], "Text", PageWidth->PaperWidth], Cell[BoxData[ \(\(\( (*Example\ 3 b*) \)\(graph4Data[\([9, 3]\)]\)\)\)], "Input", PageWidth->PaperWidth], Cell[TextData[{ "Any graph in the database can be created by ", StyleBox["graphNo[ edges, vertices, index, {isomorph}]", FontSlant->"Italic"], " and shown by ", StyleBox["ShowGraph", FontSlant->"Italic"], " or ", StyleBox["ShowLabeledGraph", FontSlant->"Italic"], "; if it is given a temporary name such as ", StyleBox["tst", FontSlant->"Italic"], " its characteristic polynomial can be checked:-" }], "Text"], Cell[BoxData[ \(\(\( (*Example\ 3 c*) \)\(ShowLabeledGraph[tst = graphNo[4, 9, 3]]; cp[tst]\)\)\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Different Graph embeddings and incantations.", "Subsection"], Cell[TextData[{ "\tMany 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 ", StyleBox["croslnk", FontSlant->"Italic"], " 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 ", StyleBox["Length[Automorphisms[..]]", FontSlant->"Italic"], " 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." }], "Text", PageWidth->PaperWidth], Cell[BoxData[ \(\(\( (*Example\ 4*) \)\(\(MK = {MK1 = graphSynthesis[ 3, {{1. , 8}, { .5, 8, 3}}], \[IndentingNewLine]MK2 = graphSynthesis[ 3, {{1. , 16}}, {{1, 5, 1}}], \[IndentingNewLine]MKp = genPetersen[8, 3], MKlcf = graphSynthesis[3, {{1. , 16}}, {{5, \(-5\)}, 6}]}; ShowGraphArray[MK];\)\(\[IndentingNewLine]\) \( (*Length[Automorphisms[MKlcf]] // Timing*) \)\)\)\)], "Input", PageWidth->PaperWidth], Cell[TextData[{ "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 ", StyleBox["CharacteristicPolynomials", FontSlant->"Italic"], " ", ":-" }], "Text", PageWidth->PaperWidth], Cell[BoxData[ \(\(\( (*Example\ 5*) \)\({regIsomorphicQ[MK1, MK2], regIsomorphicQ[MK1, MKp], regIsomorphicQ[MK1, MKlcf]}\)\)\)], "Input", PageWidth->PaperWidth] }, Closed]], Cell[CellGroupData[{ Cell["allGraph demonstration.", "Subsection"], Cell[TextData[{ StyleBox["allGraphs", FontSlant->"Italic"], " is an experimental procedure that attempts to create and identify graphs \ fitting a specification. If ", StyleBox["pr", FontSlant->"Italic"], "=1, all gra", "phs are found, otherwise only unknown graphs are found; they are put into \ ", StyleBox["results. ", FontSlant->"Italic"], "Example 6a finds all 8/4 cubical graphs with a diametral edge." }], "Text", PageWidth->PaperWidth], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{\( (*Example\ 6 a*) \), RowBox[{ RowBox[{\(pr = 1\), ";", RowBox[{"tst", "=", RowBox[{ StyleBox["allGraphs", FontColor->RGBColor[1, 0, 0]], "[", \(3, 8, 0, {{0, 4}, 1}\), "]"}]}], ";"}], "\[IndentingNewLine]", \(ShowGraphArray[results];\)}]}]], "Input", PageWidth->PaperWidth], Cell[BoxData[ \({7, "New Graphs found."}\)], "Print"] }, Open ]], Cell["\<\ 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.\ \>", "Text", PageWidth->PaperWidth], Cell[BoxData[ RowBox[{\( (*Example\ 6 b*) \), RowBox[{ RowBox[{\(pr = 0\), ";", RowBox[{ StyleBox["allGraphs", FontColor->RGBColor[1, 0, 0]], "[", \(3, 12, 4, {{2}, 1}\), "]"}], ";"}], "\n", \(ShowGraphArray[results];\)}]}]], "Input"], Cell[BoxData[{ \(i = 1; incant[\([i]\)]\), "\[IndentingNewLine]", \(toVertx[results[\([i]\)]]; hamiltonianCycle[]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Generalized LCF notation.", "Subsection"], Cell["\<\ \tLCF 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. \ \>", "Text", PageWidth->PaperWidth], Cell[BoxData[ \(\(\( (*Example\ 7 a*) \)\(pr = 1; ShowLabeledGraph[ graphSynthesis[ 3, {{1. , 18}}, {{5, \(-5\), 7, \(-7\), 5, \(-5\)}, 3}]];\)\)\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(\(\( (*Example\ 7 b*) \)\(\(ShowLabeledGraph[ new18 = graphSynthesis[ 3, {{1. , 18}}, {{5, 3, 7, \(-7\), \(-3\), \(-5\)}, 3}] /. {regGraphOptions \[Rule] {PlotLabel -> "\"}}];\ \)\[IndentingNewLine] Length[Automorphisms[new18]]\)\)\)], "Input", PageWidth->PaperWidth], Cell[TextData[{ "\tThe LCF procedure has been generalized to any value of ", StyleBox["ed", FontSlant->"Italic"], " and to only add edges if the target vertex is not full.\n\tSome 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,\[PlusMinus]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." }], "Text", PageWidth->PaperWidth], Cell["\<\ \tAny 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:-\ \>", "Text", PageWidth->PaperWidth], Cell["\<\ (*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]\ \>", "Input", PageWidth->PaperWidth, FontFamily->"Courier New"], Cell["\<\ \tA 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:-\ \>", "Text", PageWidth->PaperWidth], Cell[BoxData[ \(\(\( (*Example\ 9*) \)\(\(ShowLabeledGraph[ frLCF = graphSynthesis[ 3, {{1. , 12}}, {{\(-5\), \(-2\), \(-4\), 2, 5, 0, 2}, 1}]];\)\[IndentingNewLine] regGraphId[frLCF]\)\)\)], "Input", PageWidth->PaperWidth] }, Closed]], Cell[CellGroupData[{ Cell["Asymmetric & other graphs.", "Subsection"], Cell["\<\ \tAsymmetric 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:-\ \>", "Text", PageWidth->PaperWidth], Cell[BoxData[ \(\(\( (*Example\ 10. \ Starter\ for\ Frucht*) \)\(ShowLabeledGraph[ graphSynthesis[ 3, {{1. , 7, 1, \[Pi]/3.5}, { .6, 4, 0, .67}, {0, 1}}]];\)\)\)], "Input", PageWidth->PaperWidth], Cell["\<\ 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). 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{-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}:-\ \>", "Text"], Cell[BoxData[ \(\(\( (*Example\ 10 b*) \)\(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}}]];\)\)\)], "Input", PageWidth->PaperWidth] }, Closed]], Cell[CellGroupData[{ Cell["Hamiltonian cycles & near-Hamiltonian cycles.", "Subsection"], Cell[TextData[{ "\t", StyleBox["hamiltonianCycle[n]", FontSlant->"Italic"], " operates on ", StyleBox["vertx", FontSlant->"Italic"], " provided by ", "the current", StyleBox[" GraphSynthesis", FontSlant->"Italic"], " data or after ", StyleBox["regGraphId[tst,True] ", FontSlant->"Italic"], "if the graph was generated by ", StyleBox["subGraphs", FontSlant->"Italic"], " or a ", StyleBox["Combinatorica", FontSlant->"Italic"], " procedure", ". It is a modification of the version in [4]. It finds the first ", StyleBox["n", FontSlant->"Italic"], " hamiltonian cycles for the most recent graph, or all the non-Hamiltonian \ sequences. These can then be shown via ", StyleBox["replot[n]", FontSlant->"Italic"], ". Two are shown for the Frucht graph, the first matching that in the \ earlier example:-" }], "Text", PageWidth->PaperWidth], Cell[BoxData[ \(\(\( (*Example\ 11*) \)\(tst = graphNo[3, 12, \(-5\)]; hamiltonianCycle[5]\[IndentingNewLine] Length[vertx]\[IndentingNewLine] \(ShowGraphArray[{tst, replot[1], replot[5]}];\)\)\)\)], "Input", PageWidth->PaperWidth], Cell[TextData[{ "\tNon-Hamiltonian graphs may have many near-hamiltonian cycles, which are \ also found by ", StyleBox["hamiltonianCycle[]", FontSlant->"Italic"], ". This creates the global variable ", StyleBox["Hc", FontSlant->"Italic"], " 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.)" }], "Text", PageWidth->PaperWidth], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*Example\ 12*) \)\(ShowLabeledGraph[ Coxeter = gS[3, {{ .2, 7, 3, .175}, {1. , 7, 0}, { .5, 7, 2, .35}, { .7, \(-7\), 1, \(- .44\)}}]]; hamiltonianCycle[]\[IndentingNewLine] \(ShowGraphArray[{replot[3], replot[132, \(-1.3\)], replot[144, 2], replot[470, .2]}];\)\)\)\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ \({600, "non-Hamiltonian Circuits"}\)], "Print"], Cell[BoxData[ \({1, 8, 15, 20, 13, 6, 3, 10, 17, 19, 21, 14, 7, 4, 11, 18, 16, 9, 2, 5, 12, 26, 27, 28, 22, 23, 24, 25}\)], "Output"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .23256 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 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{-1.33604, -0.0585364, 0.0168509, 0.0168509}, {{146.75, 209.875}, {64.625, \ 1.5}} -> {-2.49308, -0.0581103, 0.0167899, 0.0168231}, {{216.188, 279.25}, \ {64.625, 1.5}} -> {-3.67481, -0.0445328, 0.0168509, 0.0164284}}] }, Open ]], Cell["\<\ Incantations can be synthesized for replots by setting up the appropriate \ circuits and adding the edges:-\ \>", "Text"], Cell[BoxData[ \(\(\( (*Example\ 12 a\ Labeled\ Target\ \ graph*) \)\(\[IndentingNewLine]\)\(ShowLabeledGraph[ replot[3]];\)\)\)], "Input"], Cell[BoxData[ \(\(\( (*Example\ 12 b\ Coxeter\ incantation\ by\ adding\ \ edges*) \)\(\[IndentingNewLine]\)\(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]\)\)\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Disconnected graphs.", "Subsection"], Cell[TextData[{ "\tDisconnected graphs can be created, but few are in the database (their \ names end in ", StyleBox["n", FontSlant->"Italic"], "x); they have no cycles, hamiltonian or otherwise:-" }], "Text", PageWidth->PaperWidth], Cell[BoxData[ \(\(\( (*Example\ 13*) \)\(ShowLabeledGraph[ tst = gS[3, {{1. , 8, 2}}, {{1, 4, \(-1\)}}]]; regGraphId[tst]\[IndentingNewLine] hamiltonianCycle[]\)\)\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(\(\( (*Example\ 13 a*) \)\(ShowGraph[ tst = gS[3, {{1. , 3, 1, .525}, {2.3, 12, 4, .52}, {1. , 3, 1, 2.1}}, {{6, 5, \(-1\)}}]]; regGraphId[tst]\[IndentingNewLine] hamiltonianCycle[]\)\)\)], "Input", PageWidth->PaperWidth] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["genPetersen", FontSlant->"Italic"], " graphs." }], "Subsection"], Cell[TextData[{ "\tgenPetersen[inner orbit size o, stellation s]", StyleBox[" creates a stellated inner orbit of the specified size ", FontSlant->"Plain"], "o", StyleBox[" linked to an outer orbit of the same size, unless the stellation \ ", FontSlant->"Plain"], "s", StyleBox[" is diametral, ", FontSlant->"Plain"], "s=o/2", StyleBox[", when the sign of the stellation determines how the graph is \ completed. If it is positive, the inner vertices link to ", FontSlant->"Plain"], StyleBox["two", FontWeight->"Bold", FontSlant->"Plain"], StyleBox[" outer vertices, so that the outer orbit has 2", FontSlant->"Plain"], "o", StyleBox[" vertices. If it is negative, the inner orbit vertices are \ pairwise joined to give each vertex three edges. If ", FontSlant->"Plain"], "o", StyleBox[" is negative and < -2, a variety of different graphs involving \ three orbits are given by some values of ", FontSlant->"Plain"], "s", StyleBox[":-", FontSlant->"Plain"] }], "Text", PageWidth->PaperWidth, FontSlant->"Italic"], Cell[BoxData[ \(\(\( (*Example\ 14*) \)\(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]];\)\)\)], "Input", PageWidth->PaperWidth] }, Closed]], Cell[CellGroupData[{ Cell["Peninsula graphs.", "Subsection"], Cell["\<\ \tPeninsula 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\".\ \>", "Text", PageWidth->PaperWidth], Cell[BoxData[ \(\(\( (*Example\ 15*) \)\(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]}}];\)\)\)], "Input", PageWidth->PaperWidth] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["subGraph", FontSlant->"Italic"], " examples." }], "Subsection"], Cell[TextData[{ "subGraph", StyleBox[" 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 ", FontSlant->"Plain"], "subGraph", StyleBox[" are in the next section. The graph is plotted if ", FontSlant->"Plain"], "depth", StyleBox[" is an integer (i.e. not if it is real, with a decimal point).", FontSlant->"Plain"], "\n", StyleBox["I have not always found pretty plots, and so have put some new \ graphs in the database in ", FontSlant->"Plain"], "subGraph", StyleBox[" form. Some include names in the Gallery.\n(Examples 16 & 17 \ deleted.)", FontSlant->"Plain"] }], "Text", FontSlant->"Italic"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Some", FontSlant->"Plain"], " of Ed Pegg's examples & other HS subgroups." }], "Subsubsection"], Cell[TextData[{ StyleBox["I had to write a clumsy search procedure to find 15 unconnected \ vertices for Ex.5 (in ", FontSlant->"Plain"], StyleBox["subGraphs ", FontSlant->"Italic"], "development). " }], "Text"], Cell[BoxData[ \(tst =. ; tst1 =. ; HS = graphNo[7, 50, 1];\)], "Input"], Cell[BoxData[ \(\(\( (*Example\ 18. \ Ed\ Pegg[5\ Ex . \ 4], \ Selecting\ 15\ unconnected\ vertices\ give\ O\_\(\(4\)\(\ \)\)*) \)\(pr \ = 0; ShowGraph[ tst = subGraph[{2, 4, 10, 13, 16, 18, 23, 25, 29, 33, 35, 39, 43, 46, 48}, 0. , {}, graphNo[7, 50, 1]]]; mx = 4; regGraphId[tst, True]\)\)\)], "Input"], Cell[BoxData[{ \(TraditionalForm\`hamiltonianCycle[ 100] (*Slow*) \), "\[IndentingNewLine]", \(TraditionalForm\`Length[Hc]\)}], "Input"], Cell[BoxData[ \(TraditionalForm\`nn = 90; ShowGraphArray[{{replot[1 + nn], replot[2 + nn]}, {replot[3 + nn], replot[4 + nn]}, {replot[5 + nn], replot[6 + nn]}, {replot[7 + nn], replot[8 + nn]}, {replot[9 + nn], replot[10 + nn]}}];\)], "Input"], Cell["\<\ The neighbours of vertex 2 are also deleted for the next example.\ \>", "Text"], Cell[BoxData[ RowBox[{\( (*Example\ 19. \ Ed\ Pegg[5\ Ex . \ 4 a], \ Deleting\ 15\ unconnected\ vertices\ and\ 7\ neighbours\ of\ one\ of\ \ them\ gives\ Coxeter*) \), RowBox[{\(pr = 0;\), "\[IndentingNewLine]", RowBox[{ RowBox[{ StyleBox["subGraph", FontColor->RGBColor[1, 0, 0]], "[", \({1, 2, 3, 4, 7, 10, 13, 15, 16, 18, 23, 25, 26, 29, 33, 35, 36, 39, 43, 45, 46, 48}, 0. , {}, graphNo[7, 50, 1]\), "]"}], ";"}], "\[IndentingNewLine]", \(regGraphId[tst1, True]\)}]}]], "Input"], Cell[TextData[{ "Using ", StyleBox["subGraph", FontSlant->"Italic"], " to delete any connected pair of Hoffman-Singleton vertices and their \ neighbours gives the Sylvester Graph:-" }], "Text"], Cell[BoxData[ RowBox[{\( (*Example\ 20. \ Ed\ Pegg[5\ Ex\ 5. ], \ Deleting\ two\ connected\ vertices\ give\ the\ Sylvester\ Graph*) \), RowBox[{\(pr = 0;\), "\[IndentingNewLine]", RowBox[{ RowBox[{ StyleBox["subGraph", FontColor->RGBColor[1, 0, 0]], "[", \({1, 2}, \(-1. \), {}, graphNo[7, 50, 1]\), "]"}], ";"}], "\[IndentingNewLine]", \(regGraphId[tst1, True]\)}]}]], "Input"], Cell[BoxData[ RowBox[{\( (*Example\ 21. \ Ed\ Pegg\ [5\ Ex\ 6. ], \[IndentingNewLine]\ GeneralizedPetersen[12, 5] . \ Vertex\ 1\ connects\ to\ {2, 5, 9, 20, 23, 33, 50}; \ choose\ any\ 3*) \), "\[IndentingNewLine]", RowBox[{ RowBox[{\(pr = 0\), ";", RowBox[{ StyleBox["subGraph", FontColor->RGBColor[1, 0, 0]], "[", \({1, 5, 23, 50}, \(-1. \), {}, graphNo[7, 50, 1]\), "]"}], ";"}], "\[IndentingNewLine]", \(regGraphId[tst1, True]\), "\[IndentingNewLine]", \(ShowLabeledGraph[ graphNo[4, 24, 2]];\)}]}]], "Input"], Cell["\<\ 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:-\ \ \>", "Text"], Cell[BoxData[{ RowBox[{\(pr = 0\), ";", \( (*Example\ 22*) \), RowBox[{ StyleBox["subGraph", FontColor->RGBColor[1, 0, 0]], "[", \({48}, \(-1\), {}, graphNo[7, 50, 1]\), "]"}], ";"}], "\[IndentingNewLine]", \(regGraphId[tst1, True]\)}], "Input"], Cell["\<\ 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)\ \>", "Text"], Cell[BoxData[ RowBox[{\( (*Example\ 23*) \), RowBox[{ RowBox[{\(pr = 0\), ";", RowBox[{ StyleBox["subGraph", FontColor->RGBColor[1, 0, 0]], "[", \({1, 2, 5}, \(-1. \), {}, graphNo[7, 50, 1]\), "]"}]}], "\[IndentingNewLine]", \(regGraphId[tst1, True]\), "\n", \(hamiltonianCycle[3];\), "\n", \(ShowLabeledGraph[replot[3]];\)}]}]], "Input"], Cell["\<\ The next graph eliminates 4 (selected) vertices and their neighbours. A \ symmetrical embedding has been put in the database.\ \>", "Text"], Cell[BoxData[ RowBox[{\( (*Example\ 24*) \), RowBox[{ RowBox[{\(pr = 0\), ";", RowBox[{ StyleBox["subGraph", FontColor->RGBColor[1, 0, 0]], "[", \({2, 4, 6, 9}, \(-1. \), {}, graphNo[7, 50, 1]\), "]"}]}], "\[IndentingNewLine]", \(regGraphId[tst1, True]\), "\[IndentingNewLine]", \(ShowLabeledGraph[ graphNo[4, 24, 2]];\)}]}]], "Input"], Cell["\<\ A random search produced the Desargues graph. The 5 specified vertices each \ share one neighbour pairwise, so 30 are rejected:-\ \>", "Text"], Cell[BoxData[ RowBox[{\( (*Example\ 25\ Desargues\ *) \), RowBox[{ RowBox[{\(pr = 0\), ";", RowBox[{ StyleBox["subGraph", FontColor->RGBColor[1, 0, 0]], "[", \({8, 13, 37, 45, 48}, \(-1. \), {}, graphNo[7, 50, 1]\), "]"}]}], "\[IndentingNewLine]", \(regGraphId[tst1, True]\), "\[IndentingNewLine]", \(ShowLabeledGraph[ graphNo[3, 20, 2]];\)}]}]], "Input"], Cell["\<\ Deleting two unconnected vertices and their neighbours gives an irregular 35 \ vertex graph with 4 vertices having 6, the rest 5, edges:-\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{\( (*Example\ 26*) \), RowBox[{ RowBox[{"tst1", "=", RowBox[{ StyleBox["subGraph", FontColor->RGBColor[1, 0, 0]], "[", \({1, 10}, \(-1\), {}, graphNo[7, 50, 1]\), "]"}]}], "\[IndentingNewLine]", \(regGraphId[tst1, True]\)}]}]], "Input", PageWidth->PaperWidth], Cell[BoxData[ \({"Not Regular, edges=", 90, "vertices=", 35}\)], "Print"] }, Open ]], Cell["\<\ None of the extra edges join pairs of vertices, so they cannot be \ cancelled.\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Other ", StyleBox["subGraph", FontSlant->"Italic"], " examples." }], "Subsubsection"], Cell[TextData[{ "These graphs were added to the database using ", StyleBox["subGraph", FontSlant->"Italic"], " incantations.\nDropping the central 4 of ", StyleBox["graphNo[4,24,1] ", FontSlant->"Italic"], "( the SmallRhombicuboctahedron, a nice example to work on) requires the \ circuit 13 to 20 to be joined. This can be done in several ways." }], "Text", PageWidth->PaperWidth], Cell[BoxData[ RowBox[{\( (*Example\ 27*) \), RowBox[{ RowBox[{ StyleBox["subGraph", FontColor->RGBColor[1, 0, 0]], "[", \({21, 22, 23, 24}, \(-1\), {{5, 8}, {6, 11}, {7, 10}, {9, 12}}, graphNo[4, 24, 1]\), "]"}], "\[IndentingNewLine]", \(regGraphId[tst1, True]\)}]}]], "Input"], Cell[BoxData[ RowBox[{\( (*Example\ 28*) \), RowBox[{ RowBox[{ StyleBox["subGraph", FontColor->RGBColor[1, 0, 0]], "[", \({21, 22, 23, 24}, \(-1\), {{5, 9}, {6, 10}, {7, 11}, {8, 12}}, graphNo[4, 24, 1]\), "]"}], "\[IndentingNewLine]", \(regGraphId[tst1, True]\)}]}]], "Input"], Cell[BoxData[ RowBox[{\( (*Example\ 29*) \), RowBox[{ RowBox[{ StyleBox["subGraph", FontColor->RGBColor[1, 0, 0]], "[", \({21, 22, 23, 24}, \(-1\), {{5, 9}, {6, 10}, {7, 12}, {8, 11}}, graphNo[4, 24, 1]\), "]"}], "\[IndentingNewLine]", \(regGraphId[tst1, True]\)}]}]], "Input"], Cell[TextData[{ "Removing the outer and inner squares and repairing symmetrically gave a \ highly symmetrical graph that was not originally in ", StyleBox["graph4Data", FontSlant->"Italic"], "." }], "Text", PageWidth->PaperWidth], Cell[BoxData[ RowBox[{\( (*Example\ 30*) \), RowBox[{ RowBox[{ RowBox[{ StyleBox["subGraph", FontColor->RGBColor[1, 0, 0]], "[", \({1, 2, 3, 4, 21, 22, 23, 24}, 0, {{5, 8}, {6, 11}, {7, 10}, {9, 12}, {13, 16}, {14, 19}, {15, 18}, {17, 20}}, graphNo[4, 24, 1]\), "]"}], ";"}], "\[IndentingNewLine]", \(regGraphId[tst1, True]\)}]}]], "Input"], Cell["\<\ Removing four vertices from the first 4,16 graph proved that I had not found \ all the 12-element graphs:-\ \>", "Text"], Cell[BoxData[ RowBox[{\( (*Example\ 31*) \), RowBox[{ RowBox[{ RowBox[{ StyleBox["subGraph", FontColor->RGBColor[1, 0, 0]], "[", \({1, 5, 9, 13}, 0, {}, graphNo[6, 16, 1]\), "]"}], ";"}], "\[IndentingNewLine]", \(regGraphId[tst1, True]\)}]}]], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "PP", StyleBox["n ", FontSlant->"Italic"], "(", StyleBox[" ", FontSlant->"Italic"], StyleBox["n ", FontSlant->"Italic"], "prime)", " examples." }], "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(pr = 0; ShowLabeledGraph[PP3 = PPn[3]]; Testpp\)], "Input"], Cell[BoxData[ \({"OK"}\)], "Print"] }, Open ]], Cell["\<\ Fig 1. Demonstration of PP3 creation and testing. {OK} means that all pairings of vertices occur in exactly one line.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(pp // tf\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "2", "3", "4", "0", "0", "0", "0", "0", "0", "0", "0", "0"}, {"1", "0", "0", "0", "5", "6", "7", "0", "0", "0", "0", "0", "0"}, {"1", "0", "0", "0", "0", "0", "0", "8", "9", "10", "0", "0", "0"}, {"1", "0", "0", "0", "0", "0", "0", "0", "0", "0", "11", "12", "13"}, {"0", "2", "0", "0", "5", "0", "0", "8", "0", "0", "11", "0", "0"}, {"0", "2", "0", "0", "0", "6", "0", "0", "9", "0", "0", "12", "0"}, {"0", "2", "0", "0", "0", "0", "7", "0", "0", "10", "0", "0", "13"}, {"0", "0", "3", "0", "5", "0", "0", "0", "9", "0", "0", "0", "13"}, {"0", "0", "3", "0", "0", "6", "0", "0", "0", "10", "11", "0", "0"}, {"0", "0", "3", "0", "0", "0", "7", "8", "0", "0", "0", "12", "0"}, {"0", "0", "0", "4", "5", "0", "0", "0", "0", "10", "0", "12", "0"}, {"0", "0", "0", "4", "0", "6", "0", "8", "0", "0", "0", "0", "13"}, {"0", "0", "0", "4", "0", "0", "7", "0", "9", "0", "11", "0", "0"} }, ColumnAlignments->{Decimal}], "\[NoBreak]", ")"}], TraditionalForm]], "Output"] }, Open ]], Cell["\<\ 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. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Transpose[pl] // tf\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "1", "1", "1", "2", "2", "2", "3", "3", "3", "4", "4", "4"}, {"2", "5", "8", "11", "5", "6", "7", "5", "6", "7", "5", "6", "7"}, {"3", "6", "9", "12", "8", "9", "10", "9", "10", "8", "10", "8", "9"}, {"4", "7", "10", "13", "11", "12", "13", "13", "11", "12", "12", "13", "11"} }, ColumnAlignments->{Decimal}], "\[NoBreak]", ")"}], TraditionalForm]], "Output"] }, Open ]], Cell[TextData[{ "Fig 3. ", StyleBox["pl", FontSlant->"Italic"], " for", " PP3. Each column is a linelist." }], "Text"], Cell[TextData[{ StyleBox["gShort", FontSlant->"Italic"], " provides the conversions (to ", StyleBox["gsh", FontSlant->"Italic"], ") ", "that allow RegGraph instructions," }], "Text"], Cell[BoxData[ \(gShort[PPn[3, {{0, .3}}]]; ShowLabeledGraph[gsh]; regGraphId[gsh]\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["PP4, PP8, PP9.", "Subsection"], Cell["\<\ 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.\ \>", "Text"], Cell[BoxData[ \(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}]]\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"pl4", "=", RowBox[{"Transpose", "[", RowBox[{"(", "\[NoBreak]", GridBox[{ {"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", 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