The Gauß-Bonnet operator of an infinite graph

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/13ZS47 - 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: infinite graph; completeness; difference operator; coboundary operator; Dirac type operator; essential self-adjointness; $ \chi-$completeness; Gauß-Bonnet 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. 2014. ⟨hal-00768827v3⟩