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The Discrete Laplacian of a 2-Simplicial Complex

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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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References

  1. Anné, C., Torki-Hamza, N.: The Gauss-Bonnet operator of an infinite graph. Anal. Math. Phys. 5(2), 137–159 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bonnefont, M., Golénia, S.: Essential spectrum and Weyl asymptotics for discrete Laplacians. arXiv:1406.5391v1 [math.SP] (2014)

  3. Baloudi, H., Golénia, S., Jeribi, A.: The adjacency matrix and the discrete Laplacian acting on forms. arXiv:1505.06109v1 [math.SP] (2015)

  4. Chernoff, P.R.: Essential self-adjointness of powers of generators of hyperbolic equations. J. Funct. Anal. 12, 401–414 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  5. Colin De Verdière, Y., Torki-Hamza, N., Truc, F.: Essential self-adjointness for combinatorial schrödinger operators II-metrically non complete graphs. Math. Phys. Anal. Geom. 14(1), 21–38 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dodziuk, J.: Elliptic operators on infinite graphs. In: analysis, geometry and topology of elliptic operators World Scientific, pp. 353–368 (2006)

  7. Elworthy, K.D., Li, X.-M.: Special Itô maps and an L 2-Hodge theory for one forms on path spaces. In: Stochastic processes, physics and geometry: new interplays, I (Leipzig, 1999), pp. 145-162. Amer. Math. Soc. (2000)

  8. Frank, L., Lenz, D., Wingert, D.: Intrinsic metrics for non-local symmetric Dirichlet forms and applications to spectral theory. J. Funct. Anal. 266, 4765–4808 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gaffney, M.: Hilbert space methods in the theory of harmonic integrals. Ann. Math. 78, 426–444 (1955)

    MathSciNet  MATH  Google Scholar 

  10. Kirchhoff, G.: Über de Aufösung der Gleichungen auf welche man bei der Untersuchen der linearen Vertheilung galvannischer Ströme gefüht wird. Ann. der Phys. und Chem. 72, 495–508 (1847)

    Google Scholar 

  11. Golénia, S., Schumacher, C.: The problem of dificiency indices for discrete Schrödinger operators on locally finite graphs. J. Math. Phys. 52(6), 1–17 (2011)

    Article  MATH  Google Scholar 

  12. Huang, X., Keller, M., Masamune, J., Wojciechowski, R.K.: A note on self-adjoint extensions of the Laplacian on weighted graphs. J. Funct. Anal. 265(8), 1556–1578 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Masamune, J.: Analysis of the Laplacian of an incomplete manifold with almost polar boundary. Rend. Mat. Appl. (7) 25(1), 109–126 (2005)

    MathSciNet  MATH  Google Scholar 

  14. Masamune, J.: A Liouville property and its application to the Laplacian of an infinite graph. Contemp. Math. 484, 103–115 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. 1. New York Academic Press, New York (1980)

    MATH  Google Scholar 

  16. Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. 2. New York Academic Press, New York (1975)

    MATH  Google Scholar 

  17. Lyons, R., Peres, Y.: Probability on trees and networks. Cambridge University Press, Cambridge (2014)

    MATH  Google Scholar 

  18. Schmüdgen, K.: Unbounded self-adjoint operators on Hilbert space, Graduate texts in mathematics 265. ⒸSpringer, Berlin (2012)

    Book  Google Scholar 

  19. Torki-Hamza, N.: Laplaciens de graphes infinis I-Graphes métriquement complets. Confluentes Math. 2(3), 333–350 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Wojciechowski, R.K.: Stochastic compactetness of graph, ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.)-City University of New York (2008)

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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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Authors and Affiliations

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

    Yassin Chebbi

  2. Unité de recherches Mathématiques et Applications, UR/13 ES 47, Faculté des Sciences de Bizerte, Université de Carthage, 7021, Bizerte, Tunisie

    Yassin Chebbi

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  1. Yassin Chebbi
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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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  • DOI: https://doi.org/10.1007/s11118-017-9659-1

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