$(4{\theta}^2,2{\theta}^2- {\theta}, {\theta}^2-{\theta},{\theta}^2-{\theta})$

-   $(4{\theta}^2,2{\theta}^2+ {\theta}, {\theta}^2+{\theta},{\theta}^2+{\theta})$

-   $(4{\theta}^2-1,2{\theta}^2, {\theta}^2,{\theta}^2)$

-   $(4{\theta}^2+4{\theta}+1,2{\theta}^2+2{\theta}, {\theta}^2+{\theta}-1,{\theta}^2+{\theta})$

-   $(4{\theta}^2+4{\theta}+2,2{\theta}^2+ {\theta}, {\theta}^2-1,{\theta}^2)$

Moreover, Chan determines the possible values of $x$ and $y$ that can occur in each of these cases.

Uniform mixing on SRGs
======================

If we suppose a strongly regular graph $X$ admits uniform mixing, then there are numbers $c$, $x$ and $y$ with $|c| = |x| = |y| = 1$ such that $$c\sqrt{n}U(t) = I + xA + y\bar{A},$$ and $c \sqrt{n} U(t)$ is a complex Hadamard matrix. Applying Chan’s results in the previous section, we obtain the following classification of primitive strongly regular graphs that admit uniform mixing. Our main result in this section follows.

\[srg\_theorem\] A primitive strongly regular graph $X$ with adjacency matrix $A$ has uniform mixing if and only if one of the following holds

-   $J-2A$ is a regular symmetric Hadamard matrix of order $4\theta^2$ with constant diagonal and positive row sum and $\theta$ is even.

-   $J- 2A-2I$ is a regular symmetric Hadamard matrix of order $4\theta^2$ with constant diagonal and positive row sum and $\theta$ is odd.

-   The Paley graph of order 9, which has parameters $(9,4,1,2)$.

To prove this theorem, we consider each possible 