)}$. Then, using Proposition \[pro:fracPart\] and Equation , we have $$C_\Gamma(\lfloor -\mu/s \rfloor, \lfloor 2-k/s \rfloor) = (t-1)((v - 2k + \lambda)t - (2s-r)(r+1)).$$ Suppose first that $\Gamma$ is co-connected. The right-hand side of Equation  is negative on the open interval $(\eta,1)$, where $\eta = (2s-r)(r+1)/(v - 2k + \lambda)$ is negative. Hence the corollary holds for $\Gamma$. On the other hand, for complete multipartite graphs we have $t = 0$, in which case the right-hand side of Equation  is negative.

The next corollary follows similarly, using the fact that the right-hand side of Equation  is negative on the open interval $(0,\eta)$, where $\eta = 1-(r^2 + r)/(v - 2k + \lambda)$.

\[cor:beatDel01\] Let $\Gamma$ be a co-connected type-II strongly regular graph with parameters $(v,k,\lambda,\mu)$ and eigenvalues $k > r \geqslant s$. Suppose that $$0< {\operatorname{frac} \left ( -k/s \right )} < 1-(r^2 + r)/(v - 2k + \lambda).$$ Then $C_\Gamma(\lfloor -\mu/s \rfloor, \lfloor 1-k/s \rfloor) < 0$.

\[rem:4.10\] We remark that if a type-II strongly regular graph satisfies the hypothesis of Corollary \[cor:beatDel01\] then its complement cannot also satisfy the hypothesis. Indeed, suppose that $\Gamma$ satisfies the hypothesis of Corollary \[cor:beatDel01\]. Since ${\operatorname{frac} \left ( -k/s \right )} > 