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2019 Orbigraphs: a graph-theoretic analog to Riemannian orbifolds
Kathleen Daly, Colin Gavin, Gabriel Montes de Oca, Diana Ochoa, Elizabeth Stanhope, Sam Stewart
Involve 12(5): 721-736 (2019). DOI: 10.2140/involve.2019.12.721

Abstract

A Riemannian orbifold is a mildly singular generalization of a Riemannian manifold that is locally modeled on n modulo the action of a finite group. Orbifolds have proven interesting in a variety of settings. Spectral geometers have examined the link between the Laplace spectrum of an orbifold and the singularities of the orbifold. One open question in this field is whether or not a singular orbifold and a manifold can be Laplace isospectral. Motivated by the connection between spectral geometry and spectral graph theory, we define a graph-theoretic analog of an orbifold called an orbigraph. We obtain results about the relationship between an orbigraph and the spectrum of its adjacency matrix. We prove that the number of singular vertices present in an orbigraph is bounded above and below by spectrally determined quantities, and show that an orbigraph with a singular point and a regular graph cannot be cospectral. We also provide a lower bound on the Cheeger constant of an orbigraph.

Citation

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Kathleen Daly. Colin Gavin. Gabriel Montes de Oca. Diana Ochoa. Elizabeth Stanhope. Sam Stewart. "Orbigraphs: a graph-theoretic analog to Riemannian orbifolds." Involve 12 (5) 721 - 736, 2019. https://doi.org/10.2140/involve.2019.12.721

Information

Received: 2 September 2017; Revised: 10 November 2018; Accepted: 1 January 2019; Published: 2019
First available in Project Euclid: 29 May 2019

zbMATH: 07140452
MathSciNet: MR3954292
Digital Object Identifier: 10.2140/involve.2019.12.721

Subjects:
Primary: 05C20‎ , 05C50
Secondary: 60J10

Keywords: Directed graph , graph spectrum , orbifold , Regular graph

Rights: Copyright © 2019 Mathematical Sciences Publishers

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Vol.12 • No. 5 • 2019
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