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The Discrete Laplacian of a 2-Simplicial Complex

Abstract

In this paper, we introduce the notion of oriented faces especially triangles in a connected oriented locally finite graph. This framework then permits to define the Laplace operator on this structure of the 2-simplicial complex. We develop the notion of χ-completeness for the graphs, based on the cut-off functions. Moreover, we study essential self-adjointness of the discrete Laplacian from the χ-completeness geometric hypothesis.

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Acknowledgments

I would like to sincerely thank my PhD advisors, Colette Anné and Nabila Torki-Hamza for helpful discussions. I am very thankful to them for all the encouragement, advice and inspiration. I take this chance to thank Matthias Keller for the fruitful discussions during my visit to Potsdam. I would also like to thank the Laboratory of Mathematics Jean Leray and the research unity (UR/13 ES 47) for their continuous support. This work was financially supported by the “PHC Utique” program of the French Ministry of Foreign Affairs and Ministry of higher education and research and the Tunisian Ministry of higher education and scientific research in the CMCU project number 13G1501 “Graphes, Géométrie et Théorie Spectrale”. Finally, I would like to thank the referee for the careful reading of my paper and the valuable comments and suggestions.

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Correspondence to Yassin Chebbi.

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Chebbi, Y. The Discrete Laplacian of a 2-Simplicial Complex. Potential Anal 49, 331–358 (2018). https://doi.org/10.1007/s11118-017-9659-1

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Keywords

  • Infinite graph
  • Difference operator
  • Laplacian on forms
  • Essential self-adjointness

Mathematics Subject Classification (2010)

  • 39A12
  • 05C63
  • 47B25
  • 05C12
  • 05C50