  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].
