Volume growth and spectrum for general graph Laplacians

TL;DR

This paper establishes estimates linking the volume growth of weighted graphs, measured with specialized metrics, to the bottom of the essential spectrum of general graph Laplacians, providing insights into spectral properties and conditions for their absence.

Contribution

It introduces new estimates connecting volume growth and spectral properties using adapted metrics, improving upon previous results for general graph Laplacians.

Findings

Derived bounds relating volume growth to the essential spectrum

Conditions for absence of the essential spectrum

Results applicable to unbounded Laplacians

Abstract

We prove estimates relating exponential or sub-exponential volume growth of weighted graphs to the bottom of the essential spectrum for general graph Laplacians. The volume growth is computed with respect to a metric adapted to the Laplacian, and use of these metrics produces better results than those obtained from consideration of the graph metric only. Conditions for absence of the essential spectrum are also discussed. These estimates are shown to be optimal or near-optimal in certain settings and apply even if the Laplacian under consideration is an unbounded operator.