Heat Kernel and Essential Spectrum of Infinite Graphs

2008/02/20 by Radosław K. Wojciechowski, Radoslaw K. Wojciechowski, Wojciechowski, Radoslaw K. · 6 citations
Computer Science · Mathematics · #Graph Theory and Algorithms #Graph theory and applications #Topological and Geometric Data Analysis #math.DG #math.SP #msc:39A12 #msc:58J35

paper · pdf · doi:10.48550/arxiv.0802.2745

18 pages, final version to appear in Indiana University Mathematics Journal

arxiv created 2008/04/24 · arxiv updated 2009/12/01

Abstract

We study the existence and uniqueness of the heat kernel on infinite, locally finite, connected graphs. For general graphs, a uniqueness criterion, shown to be optimal, is given in terms of the maximal valence on spheres about a fixed vertex. A sufficient condition for non-uniqueness is also presented. Furthermore, we give a lower bound on the bottom of the spectrum of the discrete Laplacian and use this bound to give a condition ensuring that the essential spectrum of the Laplacian is empty.

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