Spectral properties of generalized Paley graphs
Ricardo A. Podestá, Denis E. Videla
TL;DR
This work develops a unified spectral framework for generalized Paley graphs Γ(k,q) by expressing eigenvalues through cyclotomic Gaussian periods η_i^{(k,q)} and period polynomials. It delivers explicit spectra for small k (notably k ≤ 4 and certain k = 5 cases) and establishes a sharp integrality criterion: Γ(k,q) has integral spectrum iff k | (q−1)/(p−1). The authors then analyze semiprimitive GP-graphs, proving they are integral strongly regular graphs with rich distance-regular and pseudo-Latin square structure, and classify all Ramanujan GP-graphs, including eight infinite families, with special attention to k ≤ 4 and semiprimitive pairs. Overall, the results deepen the understanding of the interplay between finite-field arithmetic, Gaussian periods, and graph spectral properties with implications for expander constructions and combinatorial designs.
Abstract
We study the spectrum of generalized Paley graphs $Γ(k,q)=Cay(\mathbb{F}_q,R_k)$, 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 $Γ(k,q)$ are given by the Gaussian periods $η_{i}^{(k,q)}$ with $0\le i\le k-1$. Then, we explicitly compute the spectrum of $Γ(k,q)$ with $1\le k \le 4$ and of $Γ(5,q)$ for $p\equiv 1\pmod 5$ and $5\mid m$. Also, we characterize those GP-graphs having integral spectrum, showing that $Γ(k,q)$ 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 $Γ(k,q)$ with $1\le k \le 4$ or where $(k,q)$ is a semiprimitive pair.