a - \lfloor a \rfloor$. We are now ready to state our main results.

\[thm:beatDelsarteConf\] Let $\Gamma$ be a type-I strongly regular graph with $v$ vertices. Suppose that $$0 < {\operatorname{frac} \left ( \sqrt{v}/2 \right )} < 1/4 + ( \sqrt{v}-\sqrt{v+5/4} )/2.$$ Then $\omega(\Gamma) \leqslant \lfloor \sqrt{v} - 1 \rfloor$.

Follows from Theorem \[thm:CliqueAdjPol\] together with Corollary \[cor:beatDelsarte\] below.

For a prime $p$ satisfying $0 < {\operatorname{frac} \left ( \sqrt{p}/2 \right )} < 1/4 + ( \sqrt{p}-\sqrt{p+5/4} )/2$, we have that $\lfloor \sqrt{p} \rfloor = 2\lfloor \sqrt{p}/2 \rfloor$ is even. Furthermore, for $n := \lfloor \sqrt{p} \rfloor$, since $\lfloor \sqrt{p} \rfloor > \sqrt{p+5/4} - 1/2$, we have $n^2 + n - 1 > p$. Hence, if $\Gamma$ is a Paley graph on $p$ vertices then $\omega(\Gamma) \leqslant \lfloor \sqrt{p} - 1 \rfloor$ by Bachoc et al. [@Bach:Paley13 Theorem 2.1 (i)] (see [@Bach:Paley13 Remark 2.5]). Therefore, for type-I strongly regular graphs with $p$ (a prime) vertices, satisfying $0 < {\operatorname{frac} \left ( \sqrt{p}/2 \right )} < 1/4 + ( \sqrt{p}-\sqrt{p+5/4} )/2$, Theorem \[thm:beatDelsarteConf\] is a generalisation of Bachoc et al. [@Bach:Paley13 Theorem 2.1 (i)].

Let $g$ be a positive integer. Then $(1+4g,2g,g-1,g)$ is a feasible parameter tuple for a type-I strongly 