2019/01/11 by Kathleen Daly, Colin Gavin, Gabriel Montes de Oca +3
Computer Science · Mathematics · #Bounded function #Connection (principal bundle) #Generalization #Geometric Analysis and Curvature Flows #Homotopy and Cohomology in Algebraic Topology #Laplace operator #Orbifold #Riemannian manifold #Spectral geometry #Spectrum (functional analysis) #Topological and Geometric Data Analysis #Upper and lower bounds #math.CO #msc:05C20
paper · pdf · doi:10.2140/involve.2019.12.721
published as Involve 12 (2019) 721-736 · 21 pages, 2 figures
arxiv created 2019/01/11 · openalex created_date 2019/01/25 · openalex publication_date 2019/05/22 · arxiv updated 2019/05/29 · openalex updated_date 2026/08/05
A Riemannian orbifold is a mildly singular generalization of a Riemannian manifold that is locally modeled on [math] 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.