number of an edge-regular graph using only its parameters. Indeed, by Theorem \[thm:CliqueAdjPol\], for an edge-regular graph $\Gamma$ and some integer $c \geqslant 2$, if there exists an integer $b$ such that $C_\Gamma(b,c) < 0$ then $\omega(\Gamma) \leqslant c-1$. Hence we define the **clique adjacency bound** (CAB) to be the least integer $c \geqslant 2$ such that $C_\Gamma(b,c+1) < 0$ for some $b \in \mathbb Z$; note that such a $c$ always exists. We will show that, for a $k$-regular strongly regular graph $\Gamma$, the clique adjacency bound gives $\omega(\Gamma) \leqslant \lfloor 1 -k/s \rfloor$. That is, the clique adjacency bound is always at least as good as the Delsarte bound when applied to strongly regular graphs. This follows from Theorem \[thm:CliqueAdjPol\] together with Theorem \[thm:CAB-Del\] below. More interestingly, we will also show that the clique adjacency bound does better than the Delsarte bound for infinitely many feasible parameter tuples for strongly regular graphs. In this section we consider the univariate polynomial $C_\Gamma(f(t), g(t))$ in the variable $t$, where $f(t)$ and $g(t)$ are linear polynomials in $t$. The main idea is to choose the linear polynomials $f$ and $g$ such that there exists $t \in \R$ such that $C_\Gamma(f(t), g(t)) 