Eigenvalues of Universal Covers and the Matching Polynomial

Abstract

In this work, we prove that the universal and maximal abelian covers of a finite multi-graph have the same eigenvalues. This result strengthens a recent theorem of Li, Magee, Sabri, and Thomas (2025) and answers one of their questions. Our proof builds upon their new characterization of the point spectrum of maximal abelian covers in terms of matching polynomials. It is based on the theory of the matching polynomial and its Gallai-Edmonds decomposition.