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Conditions for the parameters of the block graph of quasi-symmetric designs

Rajendra M. Pawale
Organization: University of Mumbai
Department: Department of Mathematics
Mohan S. Shrikhande
Organization: Central Michigan University
Department: Department of Mathematics
Shubhada M. Nyayate
Organization: Dnyanasadhana College
Department: Department of Mathematics
Journal / Anthology

The electronic journal of combinatorics
Year: 2015
Volume: 22
Issue: 1

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 = 􀀀k􀀀x y􀀀x . 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.

*Mathematics > Discrete Mathematics > Combinatorics