Heat Kernel and Essential Spectrum of Infinite Graphs

TL;DR

This paper investigates the heat kernel and spectrum of infinite graphs, providing criteria for uniqueness, non-uniqueness, and conditions for the essential spectrum to be empty, advancing understanding of spectral properties of infinite graphs.

Contribution

It introduces optimal criteria for heat kernel uniqueness, bounds on the spectrum, and conditions for the essential spectrum's absence in infinite graphs.

Findings

A criterion for heat kernel uniqueness based on maximal valence.

A sufficient condition for non-uniqueness of the heat kernel.

A lower bound on the bottom of the spectrum of the Laplacian.

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.