On the existence of self-complementary and non-self-complementary strongly regular graphs with Paley parameters

For p an odd prime, let A_p be the complete classical affine association scheme whose associate classes correspond to parallel classes of lines in the classical affine plane AG(2, p). It is known that A_p is an amorphic association scheme. We investigate rank 3 fusion schemes of A_p whose basis graphs have the same parameters as the Paley graphs P(p²). In contrast to the Paley graphs, the great majority of graphs we detect are non-self-complementary and non-Schurian. In particular, existence of non-self-complementary graphs with Paley parameters is established for p≥ 17 , with an analogous existence result for non-Schurian such graphs when p≥ 11. We demonstrate that the number of self-complementary and non-self-complementary strongly regular graphs with Paley parameters grows rapidly as p→ ∞.