Intrinsic metrics on graphs: A survey

TL;DR

This survey explores how intrinsic metrics on graphs address disparities in Laplacian properties related to volume growth, covering recent advances in spectral theory, harmonic functions, and stochastic completeness.

Contribution

It provides a comprehensive overview of recent results on intrinsic metrics on graphs, highlighting their role in resolving disparities in Laplacian analysis and spectral properties.

Findings

Intrinsic metrics help unify Laplacian analysis on graphs.

Spectral bounds relate to isoperimetric constants and volume growth.

p-independence of spectra under volume growth conditions.

Abstract

A few years ago various disparities for Laplacians on graphs and manifolds were discovered. The corresponding results are mostly related to volume growth in the context of unbounded geometry. Indeed, these disparities can now be resolved by using so called intrinsic metrics instead of the combinatorial graph distance. In this article we give an introduction to this topic and survey recent results in this direction. Specifically, we cover topics such as Liouville type theorems for harmonic functions, essential selfadjointness, stochastic completeness and upper escape rates. Furthermore, we determine the spectrum as a set via solutions, discuss upper and lower spectral bounds by isoperimetric constants and volume growth and study $p$-independence of spectra under a volume growth assumption.