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Networks and Heterogeneous Media

2009, Volume 4, Issue 3: 453-468. doi: 10.3934/nhm.2009.4.453
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Isospectral infinite graphs and networks and infinite eigenvalue multiplicities

  • 1.
    LMPA Joseph Liouville, FCNRS 2956, Université du Littoral Côte d’Opale, 50, rue F. Buisson, B.P. 699, F–62228 Calais Cedex
  • 2.
    Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Jordi Girona, 1–3, 08034 Barcelona
  • Received: 01 February 2008 Revised: 01 February 2009

  • Primary: 34B45, 05C50; Secondary: 05C10, 35J05, 34L10, 35P10.

  • We consider the continuous Laplacian on infinite locally finite networks under natural transition conditions as continuity at the ramification nodes and Kirchhoff flow conditions at all vertices. It is well known that one cannot reconstruct the shape of a finite network by means of the eigenvalues of the Laplacian on it. The same is shown to hold for infinite graphs in a L∞-setting. Moreover, the occurrence of eigenvalue multiplicities with eigenspaces containing subspaces isomorphic to \l∞(\ZZ) is investigated, in particular in trees and periodic graphs.

    Citation: Joachim von Below, José A. Lubary. Isospectral infinite graphs and networks and infinite eigenvalue multiplicities[J]. Networks and Heterogeneous Media, 2009, 4(3): 453-468. doi: 10.3934/nhm.2009.4.453

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  • We consider the continuous Laplacian on infinite locally finite networks under natural transition conditions as continuity at the ramification nodes and Kirchhoff flow conditions at all vertices. It is well known that one cannot reconstruct the shape of a finite network by means of the eigenvalues of the Laplacian on it. The same is shown to hold for infinite graphs in a L∞-setting. Moreover, the occurrence of eigenvalue multiplicities with eigenspaces containing subspaces isomorphic to \l∞(\ZZ) is investigated, in particular in trees and periodic graphs.


  • This article has been cited by:

    1. Palle E. T. Jorgensen, Unbounded graph-Laplacians in energy space, and their extensions, 2012, 39, 1598-5865, 155, 10.1007/s12190-011-0518-8
    2. Joachim von Below, José A. Lubary, Baptiste Vasseur, Some Remarks on the Eigenvalue Multiplicities of the Laplacian on Infinite Locally Finite Trees, 2013, 63, 1422-6383, 1331, 10.1007/s00025-012-0271-9
    3. Joachim Kerner, Matthias Täufer, Jens Wintermayr, Robustness of Flat Bands on the Perturbed Kagome and the Perturbed Super-Kagome Lattice, 2023, 1424-0637, 10.1007/s00023-023-01399-7
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  • © 2009 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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