1 \right \rfloor \right ) = C_\Gamma(r - t, 2 + 2r - 2t),$$ which is negative since $t < 1/2 < 3/4 + ( \sqrt{v}-\sqrt{v+5/4} )/2$. Note that $t$ cannot be equal to $1/2$ since $r$ is an algebraic integer.

The next corollary follows in a similar fashion.

\[cor:beatDelsarte\] 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 $C_\Gamma \left ( \left \lfloor (\sqrt{v}-1)/2 \right \rfloor, \left \lfloor \sqrt{v} \right \rfloor \right) < 0$.

Let $t = {\operatorname{frac} \left ( r \right )} = {\operatorname{frac} \left ( (\sqrt{v}-1)/2 \right )}$. Then by our hypothesis $1/2 < t < 3/4 + ( \sqrt{v}-\sqrt{v+5/4} )/2$. Therefore we have $$C_\Gamma \left ( \left \lfloor (\sqrt{v}-1)/2  \right \rfloor, \left \lfloor \sqrt{v} \right \rfloor \right ) = C_\Gamma(r - t, 2 + 2r - 2t),$$ which is negative since $t < 3/4 + ( \sqrt{v}-\sqrt{v+5/4} )/2$.

Type-II strongly regular graphs {#sub:non_conference_strongly_regular_graph}
-------------------------------

Let $\Gamma$ be a type-II strongly regular graph with parameters $(v,k,\lambda,\mu)$. Again, in view of Proposition \[pro:fracPart\], we evaluate $C_\Gamma(x,y)$ at points of the form $(-\mu/s - t,a -k/s - t)$ for some 