Eigenvalues of Maximal Abelian Covers

2025/08/24 by Li, Wenbo, Magee, Michael, Sabri, Mostafa +1 · 2 citations
#05C50 #81Q10 #81Q35 #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2508.17332

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.

Citations

• Rare Flat Bands for Periodic Graph Operators
• Flat bands of periodic graphs
• Universality of free random variables: atoms for non-commutative rational functions
• Point Spectrum of Periodic Operators on Universal Covering Trees
• Cut-edges and regular factors in regular graphs of odd degree
• Spectral atoms of unimodular random trees
• Interlacing Families I: Bipartite Ramanujan Graphs of All Degrees
• Gallai-Edmonds Structure Theorem for Weighted Matching Polynomial

Cited by

• Sparse graphs and their Benjamini-Schramm limits: a spectral tour
• Eigenvalues of Universal Covers and the Matching Polynomial

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