arXiv 2103.094382021-03math.NTmath.CO

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

Chi Hoi Yip

备注：: 17 pages
发表信息：: Funct. Approx. Comment. Math. 66 (2022), no. 1, 119-138
DOI：: 10.7169/facm/1981

英文摘要：

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-Döman-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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