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Conditions for the parameters of the block graph of quasi-symmetric designs
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| Organization: | University of Mumbai |
| Department: | Department of Mathematics |
| Organization: | Central Michigan University |
| Department: | Department of Mathematics |
| Organization: | Dnyanasadhana College |
| Department: | Department of Mathematics |
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| The electronic journal of combinatorics |
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A quasi-symmetric design (QSD) is a 2-(v; k; ) design with intersection numbers x and y with x < y. The block graph of such a design is formed on its blocks with two distinct blocks being adjacent if they intersect in y points. It is well known that the block graph of a QSD is a strongly regular graph (SRG) with parameters (b; a; c; d) with smallest eigenvalue m = kx yx . The classi cation result of SRGs with smallest eigenvalue m, is used to prove that for a xed pair ( > 2;m > 2), there are only nitely many QSDs. This gives partial support towards Marshall Hall Jr.'s conjecture, that for a xed > 2, there exist nitely many symmetric (v; k; )-designs. We classify QSDs with m = 2 and characterize QSDs whose block graph is the complete multipartite graph with s classes of size 3. We rule out the possibility of a QSD whose block graph is the Latin square graph LSm(n) or complement of LSm(n), for m = 3; 4. SRGs with no triangles have long been studied and are of current research in- terest. The characterization of QSDs with triangle-free block graph for x = 1 and y = x+1 is obtained and the non-existence of such designs with x = 0 or > 2(x+2) or if it is a 3-design is proven. The computer algebra system Mathematica is used to nd parameters of QSDs with triangle-free block graph for 2 6 m 6 100. We also give the parameters of QSDs whose block graph parameters are (b; a; c; d) listed in Brouwer's table of SRGs.
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