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Semi-Fredholmness of the discrete Gauss-Bonnet operator

Hèla Ayadi (Université de Nantes, Laboratoire De Mathématiques Jean Leray, Unité de recherche Mathématiques et applications (UR/ES) à la faculté des sciences de Bizerte, Nantes, France)

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

Keywords: Infinite weighted graph, Discrete Gauss-Bonnet Operator, Non-parabolicity at Infinity, Closed Range Operator