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

Abstract

A quasi-symmetric design (QSD) is a 2-$(v,k,\lambda)$ 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 =-\frac{k-x}{y-x}$.The classification result of SRGs with smallest eigenvalue $-m$, is used to prove that for a fixed pair $(\lambda\ge 2,m\ge 2)$, there are only finitely many QSDs. This gives partial support towards Marshall Hall Jr.'s conjecture, that for a fixed $\lambda\ge 2$, there exist finitely many symmetric $(v, k, \lambda)$-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 $LS_m (n)$ or complement of $LS_m (n)$, 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 $\lambda > 2(x+2)$ 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 $2\le m \le 100$. We also give the parameters of QSDs whose block graph parameters are $(b,a,c,d)$ listed in Brouwer's table of SRGs.

Quasi-symmetric designs on $ 56 $ points

“…We explore known methods relying on assumed automorphism groups and enhance them sufficiently to be able to thoroughly examine the case v = 56. One new (56, 16,18) and many new (56, 16,6) QSDs are constructed, and non-existence of (56, 12,9) and (56,20,19) QSDs with certain automorphism groups is proved.…”

“…The new (56, 16,6) QSDs significantly increase the number of known symmetric (78, 22,6) designs, in which they can be embedded as residual designs. In Section 5, the developed construction techniques are applied to (56,12,9) and (56,20,19) QSDs with automorphism groups from the previous sections. It is shown that G 48 cannot be an automorphism group of these QSDs, and (56, 12, 9) QSDs with F rob 21 and some subgroups of G 48 are also eliminated.…”

“…Tables of admissible parameters of QSDs also appear in [20], organised by the associated strongly regular graphs (SRGs). Parameters from rows 47 and 52 appear in [20,Table 1], where the SRGs are known to exist.…”

“…Tables of admissible parameters of QSDs also appear in [20], organised by the associated strongly regular graphs (SRGs). Parameters from rows 47 and 52 appear in [20,Table 1], where the SRGs are known to exist. However, parameters from rows 48 to 51 are missing from [20,Table 3], where existence of the SRGs is unknown.…”

“…Parameters from rows 47 and 52 appear in [20,Table 1], where the SRGs are known to exist. However, parameters from rows 48 to 51 are missing from [20,Table 3], where existence of the SRGs is unknown.…”