Eigenvalues of Maximal Abelian Covers
Spectral Theory
2025-08-26 v1 Mathematical Physics
Combinatorics
math.MP
Abstract
We fully characterize the eigenvalues (flat bands) of the maximal abelian cover of a finite multi-graph in terms of the combinatorics of the base graph. This solves a problem of Higuchi and Nomura (2009, Problem 6.11). We use our new criterion to prove that the maximal abelian cover of any regular multi-graph has no eigenvalues, thereby proving a conjecture of (ibid., Conjecture 6.12). In an appendix, we relate our criterion for eigenvalues of the maximal abelian cover to an existing criterion for eigenvalues of the universal cover.
Keywords
Cite
@article{arxiv.2508.17332,
title = {Eigenvalues of Maximal Abelian Covers},
author = {Wenbo Li and Michael Magee and Mostafa Sabri and Joe Thomas},
journal= {arXiv preprint arXiv:2508.17332},
year = {2025}
}
Comments
13 figures