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.
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