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The Adjacency Matrix and the Discrete Laplacian Acting on Forms

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Abstract

We complete the understanding of the question of the essential self-adjoitness and non-essential self-adjointness of the discrete Laplacian acting on 1-forms. We also discuss the notion of completeness. Moreover, we study the relationship between the adjacency matrix of the line graph and the discrete Laplacian acting on 1-forms. Thanks to it, we exhibit a condition that ensures that the adjacency matrix on line graph is bounded from below and not essentially self-adjoint.

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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  Google Scholar 

  2. Aomoto, K.: Selfadjointness and limit pointness for adjacency operators on a tree. J. Analyse Math. 53, 219–232 (1989)

    Article  MathSciNet  Google Scholar 

  3. Berkolaiko, G., Kennedy, J.B., Kurasov, P., Mugnolo, D.: Edge connectivity and the spectral gap of combinatorial and quantum graphs. arXiv:1702.05264 [math.SP]

  4. Bonnefont, M., Golénia, S.: Essential spectrum and Weyl asymptotics for discrete Laplacians. Ann. Fac. Sci. Toulouse Math. 24(6), 563–624 (2015)

    Article  MathSciNet  Google Scholar 

  5. Bonnefont, M., Golénia, S., Keller, M.: Eigenvalue asymptotics for Schrödinger operators on sparse graphs. Ann. Inst. Fourier (Grenoble) 65(5), 1969–1998 (2015)

    Article  MathSciNet  Google Scholar 

  6. Breuer, J., Keller, M.: Spectral analysis of certain spherically homogeneous graphs. Oper Matrices 7(4), 825–847 (2013)

    Article  MathSciNet  Google Scholar 

  7. Chebbi, Y.: The discrete Laplacian of a 2-Simplicial complex, Potential Anal. https://doi.org/10.1007/s11118-017-9659-1 (2017)

  8. Colin de Verdière, Y.: Théorème de Kirchhoff et théorie de Hodge, Sémin. Théor. Spectr Géom., vol. 9. Saint-Martin-d’Hères, Univ. Grenoble I (1991)

    Google Scholar 

  9. de Verdière, Y.C.: Spectres de graphes, Cours Spécialisés, 4. Société Mathématique de France, Paris, 1998. viii+?114 pp. ISBN: 2-85629-068-X.

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

    Article  MathSciNet  Google Scholar 

  11. Colin De Verdière, Y., Torki-Hamza, N., Truc, F.: Essential self-adjointness for combinatorial Schrödinger operators III- Magnetic fields. Ann. Fac. Sci. Toulouse Math. (6) 20(3), 599–611 (2011)

    Article  MathSciNet  Google Scholar 

  12. Cvetkovic, D.C., Sinic, S.K.: Towards a spectral theory of graphs based on the signless Laplacian, II. Linear Algebra Appl. 432(9), 2257–2272 (2010)

    Article  MathSciNet  Google Scholar 

  13. Davidoff, G., Sarnak, P., Valette, A.: Elementary number theory, group theory, and Ramanujan graphs, London Mathematical Society Student Texts, 55, p x + 144. Cambridge University Press, Cambridge (2003). ISBN: 0-521-82426-5

    Book  Google Scholar 

  14. Dodziuk, J.: Difference equations, isoperimetric inequality and transience of certain random walks. Trans. Amer. Math. Soc. 284(2), 787–794 (1984)

    Article  MathSciNet  Google Scholar 

  15. Dodziuk, J., Karp, L.: Spectral and function theory for combinatorial Laplacians. Geometry of random motion (Ithaca, N.Y., 1987), 25–40, Contemp Math., vol. 73. Amer. Math. Soc., Providence (1988)

    Google Scholar 

  16. Doyle, P.G., Snell, J.L.: Random walks and electric networks, Carus Mathematical Monographs, 22, p xiv+?159. Mathematical Association of America, Washington (1984). ISBN: 0-88385-024-9

    Google Scholar 

  17. Evans, T.S., Lambiotte, R.: Line graphs, link partitions and overlapping communities. Phys. Rev. E. 80, 016105 (2009)

    Article  ADS  Google Scholar 

  18. Golénia, S.: Unboundedness of adjacency matrices of locally finite graphs. Lett. Math. Phys. 93, 127–140 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  19. Golénia, S.: Hardy inequality and asymptotic eigenvalue distribution for discrete Laplacians. J. Func. Anal. 266, 2662–2688 (2014)

    Article  MathSciNet  Google Scholar 

  20. Golénia, S., Schumacher, C.: The problem of deficiency indices for discrete Schrödinger operators on locally finite graphs. J. Math. Phys 52(063512), 17 (2011)

    MATH  Google Scholar 

  21. Golénia, S., Schumacher, C.: Comment on The problem of deficiency indices for discrete Schrödinger operators on locally finite graphs. J. Math. Phys. 52, 063512 (2011). J. Math. Phys. 54(6), 064101 4 pp. 2013

    Article  ADS  MathSciNet  Google Scholar 

  22. Golénia, S., Haugomat, T.: On the a.c. spectrum of the 1D discrete Dirac operator. Methods Funct. Anal. Topology 20(3), 252–273 (2014)

    MathSciNet  MATH  Google Scholar 

  23. Haeseler, S., Keller, M., Lenz, D., Wojciechowski, R.K.: Laplacian on infinite graph: Dirichlet and Neumann boundary conditions. J. Spectr. Theory 2(4), 397–432 (2012)

    Article  MathSciNet  Google Scholar 

  24. Harary, F.: Graph theory, Addison-Wesley Publishing Co., Reading, Mass.-Menlo Park Calif.-London ix+?274 pp. (1969)

  25. 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  Google Scholar 

  26. Jørgensen, P.E.T.: Essential self-adjointness of the graph-Laplacian. J. Math. Phys. 49(7), 073510,33 (2008)

    Article  MathSciNet  Google Scholar 

  27. Jørgensen, P.E.T., Pearse, E.P.J.: Spectral reciprocity and matrix representations of unbounded operators. J. Funct. Anal. 261(3), 749–776 (2011)

    Article  MathSciNet  Google Scholar 

  28. Kato, T.: Perturbation theory for linear operators Classics in Mathematics. Springer-Verlag, Berlin (1995). xxii+?619 pp. ISBN: 3-540-58661-X

    Book  Google Scholar 

  29. Keller, M., Lenz, D.: Unbounded Laplacians on graphs: basic spectral properties and the heat equation. Math. Model. Nat. Phenom. 5(4), 198–224 (2010)

    Article  MathSciNet  Google Scholar 

  30. Lim, L.H.: Hodge Laplacians on graphs, Geometry and Topology in Statistical Inference Proceedings of Symposia in Applied Mathematics, vol. 73. AMS, Providence (2015)

    Google Scholar 

  31. Masamune, J.: A Liouville property and its application to the Laplacian of an infinite graph, Spectral analysis in geometry and number theory, 103–115, Contemp. Math., 484. Amer. Math. Soc., Providence (2009)

    Google Scholar 

  32. Masson, D., McClary, W.K.: Classes of C 8 vectors and essential self-adjointness. J. Funct. Anal. 10, 19–32 (1972)

    Article  Google Scholar 

  33. Milatovic, O.: Essential self-adjointness of magnetic Schrödinger operators on locally finite graphs. Integr. Equ. Oper. Theory 71(1), 13–27 (2011)

    Article  MathSciNet  Google Scholar 

  34. Milatovic, O.: A Sears-type self-adjointness result for discrete magnetic Schrödinger operators. J. Math. Anal. Appl. 396(2), 801–809 (2012)

    Article  MathSciNet  Google Scholar 

  35. Milatovic, O., Truc, F.: Self-adjoint extensions of discrete magnetic Schrödinger operators. Ann. Henri Poincaré, 15(5), 917–936 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  36. Mohar, B., Omladic, M.: The spectrum of infinite graphs with bounded vertex degrees, Graphs, hypergraphs and applications (Eyba, 1984), 122–125 Teubner-Texte Math., vol. 73. Teubner, Leipzig (1985)

    Google Scholar 

  37. Nussbaum, A.E.: Quasi-analytic vectors. Ark. Mat. 6, 179–191 (1965)

    Article  MathSciNet  Google Scholar 

  38. Parra, D.: Spectral and scattering theory for Gauss-Bonnet operators on perturbed topological crystals. J. Math. Anal. Appl. 452(2), 792–813 (2017)

    Article  MathSciNet  Google Scholar 

  39. Reed, M., Simon, B.: Methods of modern mathematical physics tome I–IV. Academic Press (1978)

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

    Article  MathSciNet  Google Scholar 

  41. Weber, A.: Analysis of the physical Laplacian and the heat flow on a locally finite graph. J. Math. Anal. Appl. 370(1), 146–158 (2010)

    Article  MathSciNet  Google Scholar 

  42. Wojciechowski, R.: Stochastic compactetness of graph. Ph.D. thesis, City University of New York (2007)

  43. Wojciechowski, R.: Stochastically incomplete manifolds and graphs, random walks, boundaries and spectra, 163–179, Progr Probab., vol. 64. Birkhäuser/Springer Basel AG, Basel (2011)

    Google Scholar 

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Acknowledgments

SG was partially supported by the ANR project GeRaSic (ANR-13-BS01-0007-01) and SQFT (ANR-12-JS01-0008-01). HB enjoyed the hospitality of Bordeaux University when this work started. We would like to thank the anonymous referee, Colette Anné, Michel Bonnefont, Delio Mugnolo, and Nabila Torki-Hamza for useful discussions and comments on the text.

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Authors and Affiliations

  1. Faculté Des Sciences De Sfax, Route De Soukra km 3,5, B.P. 1171, 3000, Sfax, Tunisia

    Hatem Baloudi & Aref Jeribi

  2. Univ. Bordeaux, Bordeaux INP, CNRS, IMB, UMR 5251, F-33400, Talence, France

    Sylvain Golénia

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  1. Hatem Baloudi
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  2. Sylvain Golénia
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  3. Aref Jeribi
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Correspondence to Sylvain Golénia.

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Baloudi, H., Golénia, S. & Jeribi, A. The Adjacency Matrix and the Discrete Laplacian Acting on Forms. Math Phys Anal Geom 22, 9 (2019). https://doi.org/10.1007/s11040-019-9301-0

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