Networks and Heterogeneous Media (NHM)

Isospectral infinite graphs and networks and infinite eigenvalue multiplicities

Pages: 453 - 468, Volume 4, Issue 3, September 2009      doi:10.3934/nhm.2009.4.453

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Joachim von Below - LMPA Joseph Liouville, FCNRS 2956, Université du Littoral Côte d’Opale, 50, rue F. Buisson, B.P. 699, F–62228 Calais Cedex, France (email)
José A. Lubary - Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Jordi Girona, 1–3, 08034 Barcelona, Spain (email)

Abstract: 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 $L^\infty$-setting. Moreover, the occurrence of eigenvalue multiplicities with eigenspaces containing subspaces isomorphic to $\l^\infty(\ZZ)$ is investigated, in particular in trees and periodic graphs.

Keywords:  Locally finite graphs and networks, Laplacian, eigenvalue problems, adjacency and transition operators.
Mathematics Subject Classification:  Primary: 34B45, 05C50; Secondary: 05C10, 35J05, 34L10, 35P10.

Received: February 2008;      Revised: February 2009;      Available Online: July 2009.