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 strengthens a recent result of Li, Magee, Sabri, and Thomas (2025), and answers one of their questions. Our proof relies on the theory of the matching polynomial of a graph, and we further prove new results about its Gallai--Edmonds decomposition.