A note on self-adjoint extensions of the Laplacian on weighted graphs

TL;DR

This paper investigates conditions under which the Laplacian on weighted graphs has unique self-adjoint and Markovian extensions, linking graph completeness and boundary capacity to extension uniqueness.

Contribution

It provides new criteria for the uniqueness of Laplacian extensions on weighted graphs based on metric completeness and boundary capacity.

Findings

Completeness of locally finite graphs implies uniqueness of extensions.

Finite capacity of the Cauchy boundary characterizes Markovian extension uniqueness.

Results connect geometric properties of graphs with spectral extension theory.

Abstract

We study the uniqueness of self-adjoint and Markovian extensions of the Laplacian on weighted graphs. We first show that, for locally finite graphs and a certain family of metrics, completeness of the graph implies uniqueness of these extensions. Moreover, in the case when the graph is not metrically complete and the Cauchy boundary has finite capacity, we characterize the uniqueness of the Markovian extensions.