Eigenvalues of Universal Covers and the Matching Polynomial
Thomás Jung Spier
TL;DR
This work proves that the universal cover $G^{\mathrm{uni}}$ and the maximal abelian cover $G^{\mathrm{ab}}$ of any finite weighted multi-graph $G$ share the same eigenvalues. The authors develop a unifying framework based on the matching polynomial and the $\theta$-Gallai-Edmonds decomposition, connecting spectral properties of covers to combinatorial objects such as $\theta$-Aomoto subsets, 2-regular subgraphs, and molecular polynomials $M^G(x)$. They establish a broad equivalence: $\theta$ is an eigenvalue of the covers if and only if it satisfies multiple equivalent conditions involving Aomoto subsets, zeros of $\mu^{G\setminus\Gamma}$ and $\phi^{G\setminus\Gamma}$ for all 2-regular $\Gamma$, zeros of $\phi^G_\xi$ for all $\xi$, and zeros of all $M^G(x)$. When no $\theta$-Aomoto subset exists, the paper provides a constructive cycle-decomposition result: there are $k=m_\theta(G)$ disjoint cycles whose removal eliminates the $\theta$-multiplicity, enabling a precise understanding of the density of states for the universal cover. Overall, the work offers an algorithmically tractable, combinatorial lens for equating the spectra of the two canonical covers and extends Li et al.’s findings with a robust theoretical scaffold.
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.