The role of expanders in the spectral geometry of metric graphs

Delio Mugnolo

arXiv:2607.14312v2 Primary: math.SP Secondary: math.CO, math.FA

Abstract

After reviewing combinatorial and spectral definitions of expanders, we use precise lower bounds on Ramanujan graphs to investigate upper bounds -- and, specifically, the lack thereof -- on the eigenvalues of the Laplacian on \emph{metric} graphs in terms of volume, diameter, girth, mean distance, and torsional rigidity, among others.

Disclosure

“(Aveiro) and Noah Kravitz (Oxford) and Pavel Kurasov (Stockholm) for many interesting discussions. Portions of the manuscript were revised with the assistance of the LLMs Qwen, Gemini and Claude. 1”
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Structural counts

Pages 23 pdf

Theorems 9 pdf fallback

Lemmas 7 pdf fallback

Propositions 6 pdf fallback

Corollaries 2 pdf fallback

Definitions 4 pdf fallback

Displayed equations 167 pdf fallback

Bibliography entries 45 pdf fallback

Appendix pages 0 estimated

Count notes

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