Skip to main content
Log in

Unboundedness of Adjacency Matrices of Locally Finite Graphs

  • Published:
Letters in Mathematical Physics Aims and scope Submit manuscript

Abstract

Given a locally finite simple graph so that its degree is not bounded, every self-adjoint realization of the adjacency matrix is unbounded from above. In this note, we give an optimal condition to ensure it is also unbounded from below. We also consider the case of weighted graphs. We discuss the question of self-adjoint extensions and prove an optimal criterium.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Berezanski  J.M.: Expansions in Eigenfunctions of Selfadjoint Operators, ix + 809 pp. American Mathematical Society, Providence (1968)

  2. Chung, F.R.K.: Spectral graph theory. In: Regional Conference Series in Mathematics, vol. 92, xii + 207 pp. American Mathematical Society, Providence (1997)

  3. Cvetković, D., Doob, M., Sachs, H.: Spectra of graphs. Theory and application, 2nd edn, 368 pp. VEB Deutscher Verlag der Wissenschaften, Berlin (1982)

  4. Davidoff, G., Sarnak, P., Valette, A.: Elementary number theory, group theory, and Ramanujan graphs. In: London Mathematical Society Student Texts, vol. 55, x+144 pp. Cambridge University Press, Cambridge (2003)

  5. Doyle, P.G., Snell, J.L.: Random walks and electric networks. In: The Carus Mathematical Monographs, 159 pp. The Mathematical Association of America, Providence (1984)

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

    Google Scholar 

  7. Keller, M., Lenz, D.: Dirichlet forms and stochastic completeness of graphs and subgraphs (2009). arXiv:0904.2985v1 [math.FA]

  8. Keller M., Lenz D.: Unbounded Laplacians on graphs: basic spectral properties and the heat equation. Math. Model. Nat. Phenom. 5(2), 27 (2009)

    Google Scholar 

  9. Mohar, B., Omladic, M.: The spectrum of infinite graphs with bounded vertex degrees, graphs, hypergraphs and applications. In: Teubner Texte, vol. 73, pp. 122–125. Teubner, Leipzig (1985)

  10. Mohar B., Woess W.: A survey on spectra of infinite graphs. J. Bull. Lond. Math. Soc. 21(3), 209–234 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  11. Müller V.: On the spectrum of an infinite graph. Linear Algebra Appl. 93, 187–189 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  12. Reed M., Simon B.: Methods of Modern Mathematical Physics, Tome I–IV: Analysis of operators. Academic Press, London (1975)

    Google Scholar 

  13. Schulz-Baldes, H.: Geometry of Weyl theory for Jacobi matrices with matrix entries (2008). arXiv:0804.3746v1 [math-ph]

  14. Weber, A.: Analysis of the Laplacian and the heat flow on a locally finite graph (2008). arXiv:0801.0812v3 [math.SP]

  15. Wojciechowski R.: Heat kernel and essential spectrum of infinite graphs. Indiana Univ. Math. J. 58(3), 1419–1442 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  16. Wojciechowski, R.: Stochastic Completeness of Graphs. Ph.D. Thesis (2007). arXiv: 0712.1570v2[math.SP]

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Sylvain Golénia.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Golénia, S. Unboundedness of Adjacency Matrices of Locally Finite Graphs. Lett Math Phys 93, 127–140 (2010). https://doi.org/10.1007/s11005-010-0390-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11005-010-0390-8

Mathematics Subject Classification (2000)

Keywords

Navigation