Results in Mathematics

, Volume 63, Issue 3–4, pp 1331–1350 | Cite as

Some Remarks on the Eigenvalue Multiplicities of the Laplacian on Infinite Locally Finite Trees

  • Joachim von Below
  • José A. Lubary
  • Baptiste Vasseur


We consider the continuous Laplacian on an infinite uniformly locally finite network under natural transition conditions as continuity at the ramification nodes and the classical Kirchhoff flow condition at all vertices in a L-setting. The characterization of eigenvalues of infinite multiplicity for trees with finitely many boundary vertices (von Below and Lubary, Results Math 47:199–225, 2005, 8.6) is generalized to the case of infinitely many boundary vertices. Moreover, it is shown that on a tree, any eigenspace of infinite dimension contains a subspace isomorphic to \({\ell^\infty({\mathbb N})}\) . As for the zero eigenvalue, it is shown that a locally finite tree either is a Liouville space or has infinitely many linearly independent bounded harmonic functions if the edge lengths do not shrink to zero anywhere. This alternative is shown to be false on graphs containing circuits.

Mathematics Subject Classifications

34B45 05C50 05C10 35J05 34L10 35P10 


Locally finite graphs and networks Laplacian eigenvalue multiplicities harmonic functions Liouville spaces 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ali Mehmeti F.: A characterization of generalized C -notion on nets. Integral Equ. Oper. Theory 9, 753– (1986)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    von Below J.: A characteristic equation associated to an eigenvalue problem on c 2-networks. Lin. Alg. Appl. 71, 309–325 (1985)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    von Below J.: The index of a periodic graph. Results Math. 25, 198–223 (1994)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    von Below, J.: Parabolic network equations. 2nd ed. 3rd edition to appear (1995)Google Scholar
  5. 5.
    von Below, J.: Can one hear the shape of a network. In: Partial Differential Equations on Multistructures. Lecture Notes in Pure and Applied Mathematics, vol. 219, pp. 19–36. Marcel Dekker Inc., New York (2000)Google Scholar
  6. 6.
    von Below J.: An index theory for uniformly locally finite graphs. Lin. Alg. Appl. 431, 1–19 (2009)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    von Below J., Lubary J.A.: Harmonic functions on locally finite networks. Results Math. 45, 1–20 (2004)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    von Below J., Lubary J.A.: The eigenvalues of the Laplacian on locally finite networks. Results Math. 47, 199–225 (2005)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    von Below J., Lubary J.A.: The eigenvalues of the Laplacian on locally finite networks under generalized node transition. Results Math. 54, 15–39 (2009)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    von Below J., Lubary J.A.: Isospectral infinite graphs and networks and infinite eigenvalue multiplicities. Netw. Het. Media. 4, 453–468 (2009)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    von Below, J., Gensane, T.: Massé, Some spectral estimates for periodic graphs. Cahiers du LMPA Joseph Liouville, vol. 84, (1999)Google Scholar
  12. 12.
    Biggs N.L.: Algebraic graph theory. Cambridge Tracts Math. 67. Cambridge University Press, Cambridge (1967)Google Scholar
  13. 13.
    Cattaneo C.: The spectrum of the continuous Laplacian on a graph. Monatshefte für Mathematik. 124, 215–235 (1997)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Chung, F.R.K.: Spectral graph theory. In: Conference Series in Mathematics, vol. 92. AMS, Rhode Island (1997)Google Scholar
  15. 15.
    Cvetcović D.M., Doob M., Sachs H.: Spectra of graphs. Academic Press, New York (1980)Google Scholar
  16. 16.
    Diestel R.: Graph theory. Springer, Berlin (2005)MATHGoogle Scholar
  17. 17.
    Lubary J.A.: Multiplicity of solutions of second order linear differential equations on networks. Lin. Alg. Appl. 274, 301–315 (1998)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Lubary, J.A.: On the geometric and algebraic multiplicities for eigenvalue problems on graphs. In: Partial Differential Equations on Multistructures. Lecture Notes in Pure and Applied Mathematics, vol. 219, pp. 135–146. Marcel Dekker Inc., New York (2000)Google Scholar
  19. 19.
    Lubary J.A., de Solà à-Morales J.: Nonreal eigenvalues for second order differential operators on networks with circuits. J. Math. Anal. Appl. 275, 238–250 (2002)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Lumer G.: Equations de diffusion sur les réseaux infinis, Séminaire Goulaouic—Schwartz. XVIII.1–XVIII.9 (1980)Google Scholar
  21. 21.
    Mohar B., Woess W.: A survey on spectra of infinite graphs. Bull. London Math. Soc. 21, 209–234 (1989)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Nicaise S.: Spectre des réseaux topologiques finis. Bull. Sc. Math. 2e Série. 111, 401–413 (1987)MathSciNetMATHGoogle Scholar
  23. 23.
    Volkmann L.: Fundamente der Graphentheorie. Springer, Berlin (1996)MATHCrossRefGoogle Scholar
  24. 24.
    Woess W.: Random walks on infinite graphs and groups 138. Cambridge University Press, cambridge (2000)CrossRefGoogle Scholar
  25. 25.
    Wilson R.J.: Introduction to graph theory. Oliver & Boyd, Edinburgh (1972)MATHGoogle Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  • Joachim von Below
    • 1
  • José A. Lubary
    • 2
  • Baptiste Vasseur
    • 1
  1. 1.LMPA Joseph Liouville ULCO, FR CNRS Math. 2956Université Lille Nord de France ULCOCalais CedexFrance
  2. 2.Departament de Matemàtica Aplicada IIUniversitat Politècnica de CatalunyaBarcelonaSpain

Personalised recommendations