Combinatorial curvature for planar graphs
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In this paper, the authors introduced the combinatorial curvature of an infinite planar graph G corresponding to the sectional curvatures of a manifold and proved that G is hyperbolic if its curvature is negative.Abstract:
Regarding an infinite planar graph G as a discrete analogue of a noncompact simply connected Riemannian surface, we introduce the combinatorial curvature of G corresponding to the sectional curvature of a manifold. We show this curvature has the property that its negative values are bounded above by a universal negative constant. We also prove that G is hyperbolic if its curvature is negative. © 2001 John Wiley & Sons, Inc. J Graph Theory 38: 220–229, 2001read more
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Mass transportation and rough curvature bounds for discrete spaces
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Scaled Gromov hyperbolic graphs
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The essential spectrum of the Laplacian on rapidly branching tessellations
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References
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Book
Combinatorial Group Theory
Roger C. Lyndon,Paul E. Schupp +1 more
TL;DR: In this article, the authors introduce the concept of Free Products with Amalgamation (FPAM) and Small Cancellation Theory over free products with amalgamation and HNN extensions.
Book
Graph Theory: An Introductory Course
TL;DR: In an elementary text book, the reader gains an overall understanding of well-known standard results, and yet at the same time constant hints of, and guidelines into, the higher levels of the subject.
Journal ArticleDOI
Difference equations, isoperimetric inequality and transience of certain random walks
TL;DR: In this article, an analogous difference operator is studied for an arbitrary graph, and it is shown that many properties of the Laplacian in the continuous setting (e.g., the maximum principle, the Harnack inequality, and Cheeger's bound for the lowest eigenvalue) hold for this difference operator.