# Strongly regular graphs with the same parameters as Paley graph

It is known that the Paley graph $P(q)$ for $q = 5, 9, 13$ or $17$ vertices are the only strongly regular graph with the parameters as $P(q)$.

If $q \geq 25$, is the following assertion true:

There are strongly regular graphs $G$ with the same parameters as $P(q)$ such that $G\not\cong P(q)$.

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If $p$ is a prime congruent to 3 (mod 4) and $q$ is an even power of $q$, there are the Peisert graphs which are arc-transitive, self-complementary conference graphs, not isomorphic to Paley graphs.
More generally, start with the affine plane over $GF(q)$ where $q$ is odd, with point set $GF(q)\times GF(q)$. Choose a subset $P$ of $GF(q)\cup\infty$ and define $X(P)$ to be the graph with the points of the plane as vertices, with two points adjacent if they are distinct and the slope of the affine line joining them is in $P$. Then $X(P)$ is strongly regular, and if $|P|=(1+q)/2$, then $X(P)$ is a conference graph. When $q$ is large, we get many non-isomorphic graphs. (The Peisert graphs can be obtained in this way.)