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

## TL;DR

This paper explores the spectral properties of certain strongly regular graphs, demonstrating the non-existence of Delsarte cocliques in specific cases and limiting their number in others, with implications for generalized quadrangles.

## Contribution

It establishes non-existence results for Delsarte cocliques in Taylor's 2-graphs and generalized quadrangles of order (q, q^2-q), and bounds their maximum number in cases where equality holds.

## Key findings

- Delsarte cocliques do not exist in all Taylor's 2-graphs.
- No Delsarte cocliques in point graphs of generalized quadrangles of order (q, q^2-q) for infinitely many q.
- At most two Delsarte cocliques can exist in cases where spectral bounds are tight.

## Abstract

We investigate strongly regular graphs for which Hoffman's ratio bound and Cvetcovi\'{c}'s inertia bound are equal. This means that $ve^- = m^-(e^- - k)$, 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.