---
abstract: 'We determine new upper bounds for the clique numbers of strongly regular graphs in terms of their parameters. These bounds improve on the Delsarte bound for infinitely many feasible parameter tuples for strongly regular graphs, including infinitely many parameter tuples that correspond to Paley graphs.'
address:
- 'Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan'
- 'School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS, UK'
author:
- 'Gary R. W. Greaves'
- 'Leonard H. Soicher'
bibliography:
- 'sbib.bib'
title: |
    On the clique number of a\
    strongly regular graph
---

[^1]

Introduction
============

The **clique number** $\omega(\Gamma)$ of a graph $\Gamma$ is defined to be the cardinality of a clique of maximum size in $\Gamma$. For a $k$-regular strongly regular graph with smallest eigenvalue $s < 0$, Delsarte [@Del:73 Section 3.3.2] proved that $\omega(\Gamma) \leqslant \lfloor 1 -k/s \rfloor$; we refer to this bound as the **Delsarte bound**. Therefore, since one can write $s$ in terms of the parameters of $\Gamma$, one can determine the Delsarte bound knowing only the parameters $(v,k,\lambda,\mu)$ of $\Gamma$. In this paper we determine new upper bounds 