Semi-Fredholmness of the discrete Gauss-Bonnet Operator

TL;DR

This paper characterizes when the discrete Gauss-Bonnet operator on infinite weighted graphs is semi-Fredholm, extending continuous case results and providing conditions for closed range operators.

Contribution

It establishes necessary and sufficient conditions for semi-Fredholmness of the discrete Gauss-Bonnet operator on infinite graphs, extending previous continuous and discrete work.

Findings

Necessary and sufficient condition for semi-Fredholmness.

Sufficient condition for closed range of the operator.

Extension of continuous case theorems to discrete graphs.

Abstract

In the context of an infinite locally finite weighted graph, we give a necessary and sufficientcondition for semi-Fredholmness of the Gauss-Bonnet operator. This result is a discrete version of thetheorem of Gilles Carron in the continuous case [5]. In addition, using a criterion of Anghel [2], we givea sufficient condition to have an operator of Gauss-Bonnet with closed range. Finally, this work can beconsidered as an extension of the work of Colette Ann\'e and Nabila Torki-Hamza [3].