Heat Kernel and Essential Spectrum of Infinite Graphs

Details Heat Kernel and Essential Spectrum of Infinite Graphs Heat Kernel and Essential Spectrum of Infinite Graphs

2008-02-20

arxiv.org/abs/0802.2745v4

Mathematics

Spectral Theory

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

Scientific paper

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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