On the nonexistence of pseudo-generalized quadrangles

TL;DR

This paper investigates the existence of pseudo-generalized quadrangles by analyzing strongly regular graphs, improving bounds on parameters, and eliminating many potential configurations through clique and coclique analysis.

Contribution

It provides a tighter quadratic bound on parameters for pseudo-generalized quadrangles, refining previous cubic bounds and advancing understanding of their nonexistence.

Findings

Improved the bound from cubic to quadratic in parameter s for pseudo-generalized quadrangles.

Eliminated many feasible parameter sets for these structures.

Enhanced understanding of clique and coclique structures in related graphs.

Abstract

In this paper we consider the question of when a strongly regular graph with parameters $((s+1)(st+1),s(t+1),s-1,t+1)$ can exist. These parameters arise when the graph is derived from a generalized quadrangle, but there are other examples which do not arise in this manner, and we term these {\it pseudo-generalized quadrangles}. If the graph is a generalized quadrangle then $t \leq s^2$ and $s \leq t^2$, while for pseudo-generalized quadrangles we still have the former bound but not the latter. Previously, Neumaier has proved a bound for $s$ which is cubic in $t$, but we improve this to one which is quadratic. The proof involves a careful analysis of cliques and cocliques in the graph. This improved bound eliminates many potential parameter sets which were otherwise feasible.