The Gauß-Bonnet operator of an infinite graph

L'opérateur de Gauss-Bonnet sur un graphe infini

Colette Anné (1) , Nabila Torki-Hamza (2, 3)

1 LMJL - Laboratoire de Mathématiques Jean Leray
2 ISIG-K - Institut Supérieur d'Informatique et de Gestion de Kairouan
3 UR/13ES47 - Mathématiques et Applications

We propose a general condition, to ensure essential self-adjointness for the Gauß-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.

Mots clés: $ \chi-$completeness; essential self-adjointness; Dirac type operator; Gauß-Bonnet operator; infinite graph; completeness; difference operator; coboundary operator; Gauß-Bonnet operator.

Domaines: Théorie spectrale [math.SP]; Physique mathématique [math-ph]; Analyse fonctionnelle [math.FA]

Colette Anné, Nabila Torki-Hamza. The Gauß-Bonnet operator of an infinite graph. Analysis and Mathematical Physics, 2015, 5 (2), pp., 137-159. ⟨10.1007/s13324-014-0090-0⟩. ⟨hal-00768827v4⟩