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 ${ℝ}^{n}$ 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