in particular, are satisfied by infinitely many parameter tuples that correspond to Paley graphs. Our inequalities are obtained using what we call the “clique adjacency bound” (see Section \[sec:evaluating\_the\_clique\_adjacency\_polynomial\]), a bound defined by the second author [@Soi:CAB15]. We also show that the clique adjacency bound is always at most the Delsarte bound when applied to strongly regular graphs.

The paper is organised as follows. In Section \[sec:mainresults\] we state our main results and in Section \[sec:parameters\_of\_strongly\_regular\_graphs\] we state some standard identities that we will use in our proofs. Section \[sec:evaluating\_the\_clique\_adjacency\_polynomial\] contains the proofs of our main results. In Section \[sec:a\_limitation\_of\_the\_bound\] we examine the strength of the clique adjacency bound and in Section \[sec:HoffmanDelsarteCAB\] we provide an illustrative example comparing certain bounds for the clique number of an edge-regular graph that is not necessarily strongly regular. Finally, we give an appendix in which we describe our symbolic computations.

Definitions and main results {#sec:mainresults}
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A non-empty $k$-regular graph on $v$ vertices is called **edge-regular** if there exists a constant $\lambda$ such that every pair of adjacent vertices has precisely $\lambda$ common neighbours. The triple $(v,k,\lambda)$ is called the **parameter tuple** of such a graph. A **strongly regular graph** $\Gamma$ with **parameter 