Gauss sums and the maximum cliques in generalized Paley graphs of square order

Abstract

Let GP (q, d) be the d-Paley graph defined on the finite field F q . 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 maximum cliques of their corresponding generalized Paley graphs. In particular, we show that the trivial upper bound on the clique number of GP (q, d) attains if and only d | ( √ q + 1), which strengthens the previous related results by Broere-Döman-Ridley and Schneider-Silva, as well as improves the trivial upper bound on the clique number of GP (q, d) when d ∤ ( √ q + 1).