Vol. 12, No. 5, 2019

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Orbigraphs: a graph-theoretic analog to Riemannian orbifolds

Kathleen Daly, Colin Gavin, Gabriel Montes de Oca, Diana Ochoa, Elizabeth Stanhope and Sam Stewart

Vol. 12 (2019), No. 5, 721–736
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.

Keywords
graph spectrum, regular graph, directed graph, orbifold
Mathematical Subject Classification 2010
Primary: 05C50, 05C20
Secondary: 60J10
Milestones
Received: 2 September 2017
Revised: 10 November 2018
Accepted: 1 January 2019
Published: 22 May 2019

Communicated by Kenneth S. Berenhaut
Authors
Kathleen Daly
Booz Allen Hamilton
Beavercreek, OH
United States
Colin Gavin
908 Devices
Campbell, CA
United States
Gabriel Montes de Oca
Department of Mathematics
University of Oregon
Eugene, OR
United States
Diana Ochoa
Department of Mathematical Sciences
Lewis & Clark College
Portland, OR
United States
Elizabeth Stanhope
Department of Mathematical Sciences
Lewis & Clark College
Portland, OR
United States
Sam Stewart
School of Mathematics
University of Minnesota
Minneapolis, MN
United States