Exact values and improved bounds on the clique number of cyclotomic graphs

Abstract

Let $q$ be an odd power of a prime $p$, and $S\subset {\mathbb{F}}_q^{\ast }$ such that $S=-S$ and $S/S\ne {\mathbb{F}}_q^{\ast }$. We show that the clique number of the Cayley graph $\mathrm{Cay}({\mathbb{F}}_q^{+},S)$ is at most $\sqrt{|S/S|}+\sqrt{q/p}$, improving the best-known $\sqrt{q}$ upper bound for many families of such graphs substantially. Such a new bound is strongest for cyclotomic graphs and in particular, it implies the first nontrivial upper bound on the clique number of all generalized Paley graphs of non-square order, extending the work of Hanson and Pertidis. Moreover, our new bound is asymptotically sharp for an infinite family of generalized Paley graphs, and we further discover the first nontrivial family among them for which the clique number can be exactly determined. We also obtain a new lower bound on the number of directions determined by a large Cartesian product in the affine Galois plane $AG(2,q)$, which is sharp for infinite families.

arXiv comment11 pages, revised based on referee comments

References in corpus (5)

• New results on sum-product type growth over fields
• Refined Estimates Concerning Sumsets Contained in the Roots of Unity
• On the clique number of Paley graphs of prime power order
• Maximality of subfields as cliques in Cayley graphs over finite fields
• Asymptotics for the number of directions determined by $[n]\times [n]$ in ${\mathbb{F}}_p^2$

Cited by in corpus (2)

• Van Lint-MacWilliams' conjecture and maximum cliques in Cayley graphs over finite fields, II
• Intersective sets over abelian groups