A sandwich theorem for the spectrum of graph coverings

Abstract

We show that all normal covers of a fixed finite graph G sitting between the universal and maximal abelian covers share the same atomic part of their spectral measures. We give an explicit double formula for the mass of atoms which generalises and unifies known results on universal covering trees and maximal abelian covers. It also proves an extension of a conjecture in physics literature on hyperbolic lattices. Moreover, we also establish the logarithmic Hölder regularity of the continuous part of the spectral measures. Our proof combines techniques from von Neumann algebras, matching theory and the monotone labelling method. In an appendix, we prove a converse generalised Gallai-Edmonds structural theorem which might be of independent interest.

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BibTeX RIS

BibTeX

@misc{bordenave2026sandwich,
  title = {{A sandwich theorem for the spectrum of graph coverings}},
  author = {Charles Bordenave and Wenbo Li and Joe Thomas},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.36341},
  eprint = {2609.36341},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.36341},
  note = {CC BY-NC-ND 4.0},
}

RIS

TY  - UNPB
TI  - A sandwich theorem for the spectrum of graph coverings
AU  - Charles Bordenave
AU  - Wenbo Li
AU  - Joe Thomas
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2609.36341
UR  - https://arxiv.org/abs/2609.36341
AB  - We show that all normal covers of a fixed finite graph G sitting between the universal and maximal abelian covers share the same atomic part of their spectral measures. We give an explicit double formula for the mass of atoms which generalises and unifies known results on universal covering trees and maximal abelian covers. It also proves an extension of a conjecture in physics literature on hyperbolic lattices. Moreover, we also establish the logarithmic Hölder regularity of the continuous part of the spectral measures. Our proof combines techniques from von Neumann algebras, matching theory and the monotone labelling method. In an appendix, we prove a converse generalised Gallai-Edmonds structural theorem which might be of independent interest.
ER  -

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