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The Gauss-Bonnet operator of an infinite graph

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Abstract

We propose a general condition, to ensure essential self-adjointness for the Gauss-Bonnet operator \(D=d+\delta \), based on a notion of completeness as Chernoff. This gives essential self-adjointness of the Laplace operator both for functions and 1-forms on infinite graphs. This is used to extend Flanders result concerning solutions of Kirchhoff’s laws.

Résumé

Nous proposons une condition générale qui assure le caractère essentiellement auto-adjoint de l’opérateur de Gauss-Bonnet \(D=d+\delta \), basée sur une notion de complétude comme Chernoff. Comme conséquence, l’opérateur de Laplace agissant sur les fonctions et les 1-formes de graphes infinis est essentiellement auto-adjoint. Nous utilisons ce cadre pour étendre le résultat de Flanders à propos des solutions des lois de Kirchhoff.

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Acknowledgments

Part of this work was done while the author N.T-H was visiting the University of Nantes. She would like to thank the Laboratoire de Mathématiques Jean Leray (LMJL) for its hospitality. She is greatly indebted to the research unity (UR/13 Z S 47) for its continuous support. This work was supported by Grants through both Géanpyl project (FR 2962 du CNRS Mathématiques des Pays de Loire) and PHC-Utique (13 G 15-01) “Graphes, géométrie et théorie spectrale”. The authors thank Sylvain Golenia, Matthias Keller and Ognjen Milatovic for their reading with great interest and for their remarks. They would like to thank also the anonymous referee for their numerous relevant remarks and useful suggestions.

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Authors and Affiliations

  1. Laboratoire de Mathématiques Jean Leray, Faculté des Sciences, CNRS, Université de Nantes, BP 92208, 44322, Nantes, France

    Colette Anné

  2. ISIG-K, Université de Kairouan, 3100, Kairouan, Tunisia

    Nabila Torki-Hamza

  3. (UR/13 Z S 47) FSB, 7021, Bizerte, Tunisia

    Nabila Torki-Hamza

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  1. Colette Anné
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  2. Nabila Torki-Hamza
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Correspondence to Colette Anné.

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Anné, C., Torki-Hamza, N. The Gauss-Bonnet operator of an infinite graph. Anal.Math.Phys. 5, 137–159 (2015). https://doi.org/10.1007/s13324-014-0090-0

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  • DOI: https://doi.org/10.1007/s13324-014-0090-0

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