Geometry and spectrum of rapidly branching graphs

TL;DR

This paper investigates rapidly branching graphs, establishing spectral properties, eigenvalue asymptotics, and stochastic incompleteness criteria based on vertex degree growth and isoperimetric estimates.

Contribution

It provides new spectral estimates and criteria for stochastic incompleteness for rapidly branching graphs using isoperimetric and volume growth techniques.

Findings

Spectral estimates depend on vertex degree growth

Discreteness of spectrum is characterized by degree growth

New criteria for stochastic incompleteness are introduced

Abstract

We study graphs whose vertex degree tends and which are, therefore, called rapidly branching. We prove spectral estimates, discreteness of spectrum, first order eigenvalue and Weyl asymptotics solely in terms of the vertex degree growth. The underlying techniques are estimates on the isoperimetric constant. Furthermore, we give lower volume growth bounds and we provide a new criterion for stochastic incompleteness.