Extremal Peisert-type graphs without the strict-EKR property

It is known that Paley graphs of square order have the strict-EKR property, that is, all maximum cliques are canonical cliques. Peisert-type graphs are natural generalizations of Paley graphs and some of them also have the strict-EKR property. Given a prime power $q \geq 3$, we study Peisert-type graphs of order $q^2$ without the strict-EKR property and with the minimum number of edges and we call such graphs extremal. We determine number of edges in extremal graphs for each value of $q$. If $q$ is a a square or a cube, we show the uniqueness of the extremal graph and classify all maximum cliques explicitly. Moreover, when $q$ is a square, we prove that there is no Hilton-Milner type result for the extremal graph, and show the tightness of the weight-distribution bound for both non-principal eigenvalues of this graph.