Spectral properties of generalized Paley graphs

TL;DR

This paper analyzes the spectral properties of generalized Paley graphs, providing explicit spectra for certain parameters, characterizing when they are integral, and identifying conditions under which they are Ramanujan graphs.

Contribution

It offers explicit eigenvalue computations for generalized Paley graphs with small parameters and characterizes their integrality and Ramanujan properties.

Findings

Eigenvalues are given by Gaussian periods.

Explicit spectra computed for k=1 to 4, and for k=5 under specific conditions.

Identifies when generalized Paley graphs are integral and Ramanujan.

Abstract

We study the spectrum of generalized Paley graphs $\Gamma(k,q)=Cay(\mathbb{F}_q,R_k)$, undirected or not, with $R_k=\{x^k:x\in \mathbb{F}_q^*\}$ where $q=p^m$ with $p$ prime and $k\mid q-1$. We first show that the eigenvalues of $\Gamma(k,q)$ are given by the Gaussian periods $\eta_{i}^{(k,q)}$ with $0\le i\le k-1$. Then, we explicitly compute the spectrum of $\Gamma(k,q)$ with $1\le k \le 4$ and of $\Gamma(5,q)$ for $p\equiv 1\pmod 5$ and $5\mid m$. Also, we characterize those GP-graphs having integral spectrum, showing that $\Gamma(k,q)$ is integral if and only if $p$ divides $(q-1)/(p-1)$. Next, we focus on the family of semiprimitive GP-graphs. We show that they are integral strongly regular graphs (of pseudo-Latin square type). Finally, we characterize all integral Ramanujan graphs $\Gamma(k,q)$ with $1\le k \le 4$ or where $(k,q)$ is a semiprimitive pair.