Conditions for the Parameters of the Block Graph of Quasi-Symmetric Designs

A quasi-symmetric design (QSD) is a 2-(v, k, λ) 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 classification result of SRGs with smallest eigenvalue −m, is used to prove that for a fixed pair (λ > 2,m > 2), there are only finitely many QSDs. This gives partial support towards Marshall Hall Jr.’s conjecture, that for a fixed λ > 2, there exist finitely many symmetric (v, k, λ)-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 interest. 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 λ > 2(x+2) or if it is a 3-design is proven. The computer algebra system Mathematica is used to find 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.