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September  2009, 4(3): 453-468. doi: 10.3934/nhm.2009.4.453

Isospectral infinite graphs and networks and infinite eigenvalue multiplicities

Joachim von Below 1, and  José A. Lubary 2,

1. 

LMPA Joseph Liouville, FCNRS 2956, Université du Littoral Côte d’Opale, 50, rue F. Buisson, B.P. 699, F–62228 Calais Cedex, France
2. 

Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Jordi Girona, 1–3, 08034 Barcelona, Spain

Received  February 2008 Revised  February 2009 Published  July 2009

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: adjacency and transition operators., Locally finite graphs and networks, Laplacian, eigenvalue problems.
Mathematics Subject Classification: Primary: 34B45, 05C50; Secondary: 05C10, 35J05, 34L10, 35P1.
Citation: Joachim von Below, José A. Lubary. Isospectral infinite graphs and networks and infinite eigenvalue multiplicities. Networks & Heterogeneous Media, 2009, 4 (3) : 453-468. doi: 10.3934/nhm.2009.4.453
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