2]).

\[pro:comp\] Let $\Gamma$ be a connected and co-connected strongly regular graph with parameters $(v,k,\lambda,\mu)$ and eigenvalues $k > r > s$. Then $\overline \Gamma$ is strongly regular with parameters $(v,v-k-1,v-2k+\mu-2,v-2k+\lambda)$ and eigenvalues $v-k-1 > -s-1 > -r-1$.

Finally we state some straightforward bounds for the parameters of strongly regular graphs.

\[pro:SRGbound\] Let $\Gamma$ be a strongly regular graph with parameters $(v,k,\lambda,\mu)$. Then

(i) $v - 2k + \lambda \geqslant 0$ with equality if and only if $\Gamma$ is complete multipartite;

(ii) $k - \lambda - 1 \geqslant 0$ with equality if and only if $\overline \Gamma$ is complete multipartite.

The clique adjacency polynomial {#sec:evaluating_the_clique_adjacency_polynomial}
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Now we define our main tool, the clique adjacency polynomial. Given an edge-regular graph $\Gamma$ with parameters $(v,k,\lambda)$, define the **clique adjacency polynomial** $C_\Gamma(x,y)$ as $$C_\Gamma(x,y) := x(x+1)(v-y) - 2xy(k-y+1) + y(y-1)(\lambda-y+2).$$

The utility of the clique adjacency polynomial follows from [@Soi:10 Theorem 1.1] (see also [@Soi:CAB15 Theorem 3.1]), giving:

\[thm:CliqueAdjPol\] Let $\Gamma$ be an edge-regular graph with parameters $(v,k,\lambda)$. Suppose that $\Gamma$ has a clique of size $c \geqslant 2$. Then $C_\Gamma(b,c) \geqslant 0$ for all integers $b$.

As discussed in [@Soi:10] and [@Soi:CAB15], Theorem \[thm:CliqueAdjPol\] provides a way of bounding the clique 