vertices to which we can apply either Theorem \[thm:beatDelsarteConf\] or Theorem \[thm:beatDelsarteNonConf\].[]{data-label="tab:bd"}

Feasible parameters for which there does not exist a corresponding strongly regular graph whose clique number is equal to the clique adjacency bound include $(16,10,6,6)$ and $(27, 16, 10, 8)$. However, we ask the following question. Do there exist strongly regular graphs with parameters $(v,k,\lambda,\mu)$, with $k < v/2$, such that every strongly regular graph having those parameters has clique number less than the clique adjacency bound?

Hoffman bound vs Delsarte bound vs clique adjacency bound {#sec:HoffmanDelsarteCAB}
=========================================================

Let $\Gamma$ be a connected non-complete regular graph with $v$ vertices, valency $k$, and second largest eigenvalue $r<k$. Then the complement $\overline\Gamma$ of $\Gamma$ is a regular graph with valency $\overline k=v-k-1$ and least eigenvalue $\overline s=-r-1<0$. We may obtain a bound for the clique number of $\Gamma$ by applying the Hoffman bound (also called the ratio bound) [@GM16 Theorem 2.4.1] on the size of a largest independent set (coclique) of $\overline\Gamma$. This gives $$\omega(\Gamma)\leqslant \left \lfloor \frac{v}{1-\overline{k}/\overline{s}} \right \rfloor.$$ If $\Gamma$ is strongly regular, then it is known (and follows from the relations of Proposition \[pro:SRG\]) that the Delsarte bound for $\omega(\Gamma)$ is the same as that given by 