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It is known that Paley graphs of square order have the strict-EKR property.Peisert-type graphs are natural generalizations of Paley graphs and some ofthem also have the strict-EKR property. Given a prime power $q$, we studyextremal Peisert-type graphs of order $q^2$ without the strict-EKR property,that is, Peisert-type graphs of order $q^2$ without the strict-EKR property andwith the minimum number of edges. First, we determine this minimum number ofedges for each value of $q$. If $q$ is a square, we show the uniqueness of theextremal graph and its isomorphism with certain affine polar graph. Using theisomorphism, we conclude that there is no Hilton-Milner type result for thisextremal graph. We also prove the tightness of the weight-distribution boundfor both non-principal eigenvalues of this graph. If $q$ is a cube but not asquare, we show the uniqueness of the extremal graph and determine the numberand the structure of non-canonical cliques. Finally, we show such uniquenessresult does not extend to all $q$.

Extremal Peisert-type graphs without the strict-EKR property