only holds for type-I strongly regular graphs on $5$ vertices or $9$ vertices (where we have equality). One can explicitly compute the clique adjacency bound for these two cases: the unique $(5,2,0,1)$-strongly regular graph and the unique $(9,4,1,2)$-strongly regular graph. For each of these graphs the clique adjacency bound is equal to $\lambda+2$.

Now we give a couple of remarks about Theorem \[thm:limitation\].

For strongly regular graphs with $\lambda \leqslant 1$, it is easy to see that the clique number is $\lambda + 2$. By Theorem \[thm:limitation\], the clique adjacency bound is equal to the clique number for such graphs. Let $\Gamma$ be a strongly regular graph with parameters $(v,k,\lambda,\mu)$. By Proposition \[pro:SRG\], we see that $k = -s(r+1)- r +\lambda$. Therefore, for strongly regular graphs with $\lambda=2$ and $r \geqslant 2$, we have $\lambda+1 = 3 \leqslant -k/s$, and so Theorem \[thm:limitation\] applies to such graphs.

We conjecture that if the clique adjacency bound is less than $-k/s$ then $\lambda+1 \leqslant -k/s$. We have verified this conjecture for all feasible parameter tuples for strongly regular graphs on up to $1300$ vertices, making use of Brouwer’s website [@Brouwer:URL].

In Table \[tab:bd\], we list all the feasible parameter tuples for strongly 