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

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, 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.
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
[1]

Kanishka Perera, Andrzej Szulkin. p-Laplacian problems where the nonlinearity crosses an eigenvalue. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 743-753. doi: 10.3934/dcds.2005.13.743

[2]

Qun Lin, Hehu Xie. Recent results on lower bounds of eigenvalue problems by nonconforming finite element methods. Inverse Problems & Imaging, 2013, 7 (3) : 795-811. doi: 10.3934/ipi.2013.7.795

[3]

J. Ángel Cid, Pedro J. Torres. Solvability for some boundary value problems with $\phi$-Laplacian operators. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 727-732. doi: 10.3934/dcds.2009.23.727

[4]

Yaiza Canzani, A. Rod Gover, Dmitry Jakobson, Raphaël Ponge. Nullspaces of conformally invariant operators. Applications to $\boldsymbol{Q_k}$-curvature. Electronic Research Announcements, 2013, 20: 43-50. doi: 10.3934/era.2013.20.43

[5]

Alberto Cabada, J. Ángel Cid. Heteroclinic solutions for non-autonomous boundary value problems with singular $\Phi$-Laplacian operators. Conference Publications, 2009, 2009 (Special) : 118-122. doi: 10.3934/proc.2009.2009.118

[6]

L. Yu. Glebsky and E. I. Gordon. On approximation of locally compact groups by finite algebraic systems. Electronic Research Announcements, 2004, 10: 21-28.

[7]

Ravi P. Agarwal, Kanishka Perera, Zhitao Zhang. On some nonlocal eigenvalue problems. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 707-714. doi: 10.3934/dcdss.2012.5.707

[8]

Maria Fărcăşeanu, Mihai Mihăilescu, Denisa Stancu-Dumitru. Perturbed fractional eigenvalue problems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6243-6255. doi: 10.3934/dcds.2017270

[9]

Giuseppina Barletta, Roberto Livrea, Nikolaos S. Papageorgiou. A nonlinear eigenvalue problem for the periodic scalar $p$-Laplacian. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1075-1086. doi: 10.3934/cpaa.2014.13.1075

[10]

Isabeau Birindelli, Stefania Patrizi. A Neumann eigenvalue problem for fully nonlinear operators. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 845-863. doi: 10.3934/dcds.2010.28.845

[11]

Jean-Michel Rakotoson. Generalized eigenvalue problem for totally discontinuous operators. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 343-373. doi: 10.3934/dcds.2010.28.343

[12]

M. DeDeo, M. Martínez, A. Medrano, M. Minei, H. Stark, A. Terras. Spectra of Heisenberg graphs over finite rings. Conference Publications, 2003, 2003 (Special) : 213-222. doi: 10.3934/proc.2003.2003.213

[13]

Richard Miles, Michael Björklund. Entropy range problems and actions of locally normal groups. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 981-989. doi: 10.3934/dcds.2009.25.981

[14]

Yutong Chen, Jiabao Su. Resonant problems for fractional Laplacian. Communications on Pure & Applied Analysis, 2017, 16 (1) : 163-188. doi: 10.3934/cpaa.2017008

[15]

Sarah Constantin, Robert S. Strichartz, Miles Wheeler. Analysis of the Laplacian and spectral operators on the Vicsek set. Communications on Pure & Applied Analysis, 2011, 10 (1) : 1-44. doi: 10.3934/cpaa.2011.10.1

[16]

Delio Mugnolo. Dynamical systems associated with adjacency matrices. Discrete & Continuous Dynamical Systems - B, 2018, 23 (5) : 1945-1973. doi: 10.3934/dcdsb.2018190

[17]

Julián Fernández Bonder, Leandro M. Del Pezzo. An optimization problem for the first eigenvalue of the $p-$Laplacian plus a potential. Communications on Pure & Applied Analysis, 2006, 5 (4) : 675-690. doi: 10.3934/cpaa.2006.5.675

[18]

Regino Criado, Julio Flores, Alejandro J. García del Amo, Miguel Romance. Structural properties of the line-graphs associated to directed networks. Networks & Heterogeneous Media, 2012, 7 (3) : 373-384. doi: 10.3934/nhm.2012.7.373

[19]

Isabeau Birindelli, Francoise Demengel. Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators. Communications on Pure & Applied Analysis, 2007, 6 (2) : 335-366. doi: 10.3934/cpaa.2007.6.335

[20]

J. R. L. Webb. Uniqueness of the principal eigenvalue in nonlocal boundary value problems. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 177-186. doi: 10.3934/dcdss.2008.1.177

2016 Impact Factor: 1.2

Metrics

  • PDF downloads (1)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]