On the eigenvalues of cyclic covers of Paley graphs

TL;DR

This paper investigates the eigenvalues of cyclic covers of Paley graphs, revealing conditions under which eigenvalues determine graph isomorphism and providing examples of cospectral non-isomorphic covers.

Contribution

It characterizes when eigenvalues determine isomorphism in prime field cases and constructs cospectral non-isomorphic examples for extension fields.

Findings

Eigenvalues determine isomorphism for q=p in translation-invariant covers.

Constructed cospectral non-isomorphic covers for q=p^r > p.

Provided new insights into spectral properties of cyclic covers of Paley graphs.

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.