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arXiv · 2103.09438

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

Chi Hoi Yip
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Abstract

Let $GP(q,d)$ be the $d$-Paley graph defined on the finite field $\mathbb{F}_q$. It is notoriously difficult to improve the trivial upper bound $\sqrt{q}$ 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 $d \mid (\sqrt{q}+1)$, which strengthens the previous related results by Broere-D\"oman-Ridley and Schneider-Silva. We also obtain a new simple proof of Stickelberger's theorem on evaluating semi-primitive Gauss sums.

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Chi Hoi Yip. 2021-03-17. Gauss sums and the maximum cliques in generalized Paley graphs of square order. https://doi.org/10.7169/facm%2F1981

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