The Gau{\ss}-Bonnet operator of an infinite graph

TL;DR

This paper establishes conditions for the essential self-adjointness of the Gauß-Bonnet operator on infinite graphs, extending classical results and ensuring well-posedness of related differential operators.

Contribution

It introduces a general completeness condition ensuring self-adjointness of the Gauß-Bonnet operator on infinite graphs, extending prior results to functions and 1-forms.

Findings

Essential self-adjointness of the Laplace operator on infinite graphs.

Extension of Kirchhoff's laws solutions to broader graph classes.

New completeness condition based on Chernoff's notion.

Abstract

We propose a general condition, to ensure essential self-adjointness for the Gau{\ss}-Bonnet operator, based on a notion of completeness as Chernoff. This gives essential self-adjointness of the Laplace operator both for functions or 1-forms on infinite graphs. This is used to extend Flanders result concerning solutions of Kirchhoff's laws.