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 - Institut Supérieur d'informatique et de gestion [Kairouan]
3 05/UR/15-02 - Mathématiques et Applications

We propose a general condition for selfadjointness of the Gauß-Bonnet operator D=d+delta based on the notion of negligible boundary introduced by Gaffney. This gives 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 laws.

Mots clés: essential self-adjointness; infinite graph; completeness; difference operator; coboundary operator; Dirac type operator; essential self-adjointness.

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. 2012. ⟨hal-00768827v1⟩