Unbounded Laplacians on Graphs: Basic Spectral Properties and the Heat Equation

TL;DR

This paper studies unbounded Laplacians on graphs within the framework of regular Dirichlet forms, exploring their spectral properties, selfadjointness issues, spectrum absence, and stochastic incompleteness.

Contribution

It provides a detailed analysis of unbounded graph Laplacians, highlighting phenomena like essential selfadjointness failure and spectral properties, which are less understood in this context.

Findings

Unbounded Laplacians can lack essential selfadjointness.

These operators may have no essential spectrum.

The study reveals conditions leading to stochastic incompleteness.

Abstract

We discuss Laplacians on graphs in a framework of regular Dirichlet forms. We focus on phenomena related to unboundedness of the Laplacians. This includes (failure of) essential selfadjointness, absence of essential spectrum and stochastic incompleteness.