Parametric restrictions on quasi-symmetric designs

TL;DR

This paper establishes new parametric restrictions on quasi-symmetric 2-designs based on invariants of their connected block graphs, extending classical theorems and providing explicit conditions for specific graph classes.

Contribution

It introduces new invariants for strongly regular graphs and derives parametric restrictions on quasi-symmetric 2-designs using these invariants, generalizing classical results.

Findings

Derived explicit restrictions for designs with block graphs in specific classes

Introduced discriminant and p-adic invariants for strongly regular graphs

Extended classical theorems to broader classes of designs and graphs

Abstract

In this paper, we attach several new invariants to connected strongly regular graphs (excepting conference graphs on non-square number of vertices) : one invariant called the discriminant, and a p-adic invariant corresponding to each prime number p. We prove parametric restrictions on quasi-symmetric 2-designs with a given connected block graph $G$ and a given defect (absolute difference of the two intersection numbers) solely in terms of the defect and the parameters of $G$, including these new invariants. This is a natural analogue of Schutzenberger's Theorem and the Shrikhande-Chowla-Ryser theorem. This theorem is effective when these graph invariants can be explicitly computed. We do this for complete multipartite graphs, co-triangular graphs, symplectic non-orthogonality graphs (over the field of order $2$) and the Steiner graphs, yielding explicit restrictions on the parameters of quasi-symmetric 2-designs whose block graphs belong to any of these four classes.