The Adjacency Matrix and the Discrete Laplacian Acting on Forms

  • Hatem Baloudi
  • Sylvain GoléniaEmail author
  • Aref Jeribi


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.


Discrete Laplacian Locally finite graphs Self-adjoint extension Adjacency matrix Forms 

Mathematics Subject Classification (2010)

81Q35 47B25 05C63 



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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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Faculté Des Sciences De SfaxSfaxTunisia
  2. 2.Univ. Bordeaux, Bordeaux INP, CNRS, IMB, UMR 5251TalenceFrance

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