Gauss sums and the maximum cliquesin generalized Paley graphs of square order

Let GP (q, d) be the d-Paley graph defined on the finite field Fq . It is notoriously difficult to improve the trivial upper bound √ q on the clique number of GP (q, d). In this paper, we investigate the connection between Gauss sums over a finite field and the maximum cliques of their corresponding generalized Paley graphs. We show that the trivial upper bound on the clique number of GP (q, d) is tight if and only if d | (√q + 1), which strengthens the previous related results by Broere-Döman-Ridley and Schneider-Silva. We also obtain a new simple proof of Stickelberger’s theorem on evaluating semi-primitive Gauss sums.