Ovoids of Generalized Quadrangles of Order $(q,q^2-q)$ and Delsarte Cocliques in Related Strongly Regular Graphs

Abstract

We investigate strongly regular graphs for which Hoffman's ratio bound and Cvetcović's inertia bound are equal. This means that $v{e}^{-}=m^{-}(e^{-}-k)$, where $v$ is the number of vertices, $k$ is the regularity, $e^{-}$ is the smallest eigenvalue, and $m^{-}$ is the multiplicity of $e^{-}$. We show that Delsarte cocliques do not exist for all Taylor's $2$-graphs and for point graphs of generalized quadrangles of order $(q,q^2-q)$ for infinitely many $q$. For cases where equality may hold, we show that for nearly all parameter sets, there are at most two Delsarte cocliques.

11 pages, accepted by the Journal of Combinatorial Designs plus several minor corrections