Eigenvalues of Maximal Abelian Covers

Wenbo Li; Michael Magee; Mostafa Sabri; Joe Thomas

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arXiv · 2508.17332

Eigenvalues of Maximal Abelian Covers

Wenbo Li·Michael Magee·Mostafa Sabri·Joe Thomas

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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.

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BibTeX RIS

Wenbo Li, Michael Magee, Mostafa Sabri, Joe Thomas. 2025-08-24. Eigenvalues of Maximal Abelian Covers. https://arxiv.org/abs/2508.17332

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