The nature of the spectrum of generalized Paley graphs and weak Waring numbers over finite fields

TL;DR

This paper investigates the spectral properties of generalized Paley graphs over finite fields, characterizes when they are real or integral, and applies these findings to compute weak Waring numbers.

Contribution

It provides a complete characterization of real spectrum in GP-graphs, constructs infinite families of integral GP-graphs, and links spectral properties to weak Waring numbers over finite fields.

Findings

GP-graphs are real if and only if they are undirected.

Constructed infinite families of integral GP-graphs using cyclotomic polynomials.

Reduced weak Waring number computations to classic Waring numbers over finite fields.

Abstract

We consider the family of generalized Paley graphs (GP-graphs for short) $\Gamma(k,q) = Cay(\mathbb{F}_q, (\mathbb{F}_q^*)^k)$, with $q=p^m$ and $p$ prime. We characterize all GP-graphs having real spectrum; namely, $Spec(\Gamma(k,q)) \subset \mathbb{R}$ if and only if $\Gamma(k,q)$ is undirected. We then study conditions for integrality in the spectrum and give a general method to produce integral GP-graphs through cyclotomic polynomials. Using this, we construct several infinite families of integral GP-graphs. Next, we focus on directed GP-graphs (GP-digraphs). We show that GP-digraphs always have three or more eigenvalues, and then we prove that there is only one kind of GP-digraphs having three different eigenvalues: the oriented Paley graphs $\vec{\mathcal{P}}_q$ or disjoint unions of copies of them, $\vec{\mathcal{P}}_q \cup \cdots \cup \vec{\mathcal{P}}_q$. Then, we show that generically the GP-digraphs have period 1 (equivalently index of imprimitivity 1) except for $\Gamma(q-1,q)$ with $q$ odd, which is the disjoint union of oriented $p$-cycles, having period $p$. Finally, as an application, we study weak Waring numbers over finite fields through GP-graphs. In particular, we reduce the computation of the weak Waring numbers over finite fields to the computation of classic Waring numbers over finite fields, a result previously obtained by Cochrane and Cipra in 2012 by other means.