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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
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Abstract |
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A Riemannian orbifold is a mildly singular generalization of a Riemannian manifold that is locally modeled on 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
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Mathematical Subject Classification 2010
Primary: 05C50, 05C20
Secondary: 60J10
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Milestones
Received: 2 September 2017
Revised: 10 November 2018
Accepted: 1 January 2019
Published: 22 May 2019
Communicated by Kenneth S. Berenhaut
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Authors |
| Kathleen Daly | |
| Booz Allen Hamilton Beavercreek, OH United States |
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| Colin Gavin | |
| 908 Devices Campbell, CA United States |
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| Gabriel Montes de Oca | |
| Department of Mathematics University of Oregon Eugene, OR United States |
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| Diana Ochoa | |
| Department of Mathematical
Sciences Lewis & Clark College Portland, OR United States |
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| Elizabeth Stanhope | |
| Department of Mathematical
Sciences Lewis & Clark College Portland, OR United States |
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| Sam Stewart | |
| School of Mathematics University of Minnesota Minneapolis, MN United States |
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