2]).

\[pro:SRG\] Let $\Gamma$ be a strongly regular graph with parameters $(v,k,\lambda,\mu)$ and eigenvalues $k > r \geqslant s$. Then $$\begin{aligned}
        (v-k-1)\mu &= k(k-\lambda-1); \\
        \lambda-\mu &= r+s; \\
        \mu - k &= rs.
    \end{aligned}$$

\[pro:SRGtI\] Let $\Gamma$ be a type-I strongly regular graph with parameters $(v,k,\lambda,\mu)$ and eigenvalues $k > r > s$. Then $$\begin{aligned}
        k = (v-1)/2; \quad \lambda = (v-&5)/4; \quad \mu = (v-1)/4; \\
        r = (\sqrt{v}-1)/2; \quad\quad\quad & \quad\quad s = -(\sqrt{v}+1)/2.
    \end{aligned}$$

The next proposition is a key observation.

\[pro:fracPart\] Let $\Gamma$ be a strongly regular graph with parameters $(v,k,\lambda,\mu)$ and eigenvalues $k > r \geqslant s$.

(i) If $\Gamma$ is type I then $k/s - 2{\operatorname{frac} \left ( \mu/s \right )}$ is an integer.

(ii) If $\Gamma$ is type II then $k/s-{\operatorname{frac} \left ( \mu/s \right )}$ is an integer.

If $\Gamma$ is type I then, by Proposition \[pro:SRGtI\], we have $k-2\mu = 0$. If $\Gamma$ is type II then, by Proposition \[pro:SRG\], we have $k/s-\mu/s = -r$ and $r$ is an integer.

Next, the complement $\overline \Gamma$ of a strongly regular graph $\Gamma$ is also a strongly regular graph. This is again a standard result (see Cameron and Van Lint [@CvL Chapter 