Heat Kernel and Essential Spectrum of Infinite Graphs
Spectral Theory
2008-04-24 v4 Differential Geometry
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.
Cite
@article{arxiv.0802.2745,
title = {Heat Kernel and Essential Spectrum of Infinite Graphs},
author = {Radoslaw K. Wojciechowski},
journal= {arXiv preprint arXiv:0802.2745},
year = {2008}
}
Comments
18 pages, final version to appear in Indiana University Mathematics Journal