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On the eigenvalues of cyclic covers of Paley graphs

Natalie Dinin, John A. Lind

TL;DR

The paper investigates eigenvalues of translation-invariant cyclic covers $X^{\alpha}$ of Paley graphs $X(\mathbf{F}_q)$, deriving explicit spectral formulas $\theta^{\alpha}_{a,k} = \sum_{s\in\mathbf{F}_q^{\boxtimes}} \zeta_p^{\mathrm{tr}(as)} \zeta_{\ell}^{k\alpha(s)}$. It proves a rigidity result: when $q=p$ is prime, adjacency spectra determine the isomorphism class among such covers, using Babai’s CI framework and Carlitz’s automorphism description; however, for $q=p^r$ with $r\ge 2$ there exist cospectral non-isomorphic covers, shown via odd permutation polynomials and twisted adjacency matrices. The authors present a concrete $q=25$ example and outline a general construction method, including a broader claim that cospectral non-isomorphic covers exist for many $q=p^r$ with $r>1$ (excluding $q=9$). These results link spectral data to covering graph isomorphism and illustrate limits of the “hearing the shape” paradigm in the cyclic-cover Paley graph setting.

Abstract

We study covering graphs of the Paley graph associated to a finite field of characteristic p in the case where the covering transformation group is cyclic of prime order distinct from p. When the field has q = p elements, we show that the eigenvalues of the adjacency matrix determine the graph isomorphism class among translation invariant covers. When q = p^r > p, we construct examples of cospectral covering graphs that are not isomorphic as graphs.

On the eigenvalues of cyclic covers of Paley graphs

TL;DR

The paper investigates eigenvalues of translation-invariant cyclic covers of Paley graphs , deriving explicit spectral formulas . It proves a rigidity result: when is prime, adjacency spectra determine the isomorphism class among such covers, using Babai’s CI framework and Carlitz’s automorphism description; however, for with there exist cospectral non-isomorphic covers, shown via odd permutation polynomials and twisted adjacency matrices. The authors present a concrete example and outline a general construction method, including a broader claim that cospectral non-isomorphic covers exist for many with (excluding ). These results link spectral data to covering graph isomorphism and illustrate limits of the “hearing the shape” paradigm in the cyclic-cover Paley graph setting.

Abstract

We study covering graphs of the Paley graph associated to a finite field of characteristic p in the case where the covering transformation group is cyclic of prime order distinct from p. When the field has q = p elements, we show that the eigenvalues of the adjacency matrix determine the graph isomorphism class among translation invariant covers. When q = p^r > p, we construct examples of cospectral covering graphs that are not isomorphic as graphs.
Paper Structure (5 sections, 9 theorems, 45 equations, 1 figure)

This paper contains 5 sections, 9 theorems, 45 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Paley graphs and their adjacency eigenvalues
  3. Cyclic covers of Paley graphs and their adjacency eigenvalues
  4. The Cayley isomorphism property for $X^\alpha$
  5. The construction of cospectral non-isomorphic covering graphs

Key Result

Theorem 1

Suppose that $p$ is a prime number, and that $X^{\alpha}$ and $X^{\beta}$ are translation invariant $\mathbf{Z}/\ell$-covers of the Paley graph $X(\mathbf{F}_p)$ for a prime $\ell \neq p$. If the adjacency spectra of $X^\alpha$ and $X^\beta$ are equal as multisets, then there is an isomorphism of gr

Figures (1)

  • Figure 1: A minimal example of cospectral nonisomorphic covers ($q = 25$, $\ell = 3$)

Theorems & Definitions (16)

  • Theorem 1
  • Theorem 2
  • Lemma 3.3
  • Proposition 3.4
  • proof
  • Remark 3.5
  • Theorem 3.6
  • proof
  • Theorem 4.1
  • Proposition 4.2
  • ...and 6 more