September  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]

Matthieu Alfaro, Isabeau Birindelli. Evolution equations involving nonlinear truncated Laplacian operators. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3057-3073. doi: 10.3934/dcds.2020046

[2]

Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261

[3]

Shuang Liu, Yuan Lou. A functional approach towards eigenvalue problems associated with incompressible flow. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3715-3736. doi: 10.3934/dcds.2020028

[4]

Anton A. Kutsenko. Isomorphism between one-dimensional and multidimensional finite difference operators. Communications on Pure & Applied Analysis, 2021, 20 (1) : 359-368. doi: 10.3934/cpaa.2020270

[5]

Bin Wang, Lin Mu. Viscosity robust weak Galerkin finite element methods for Stokes problems. Electronic Research Archive, 2021, 29 (1) : 1881-1895. doi: 10.3934/era.2020096

[6]

Junkee Jeon. Finite horizon portfolio selection problems with stochastic borrowing constraints. Journal of Industrial & Management Optimization, 2021, 17 (2) : 733-763. doi: 10.3934/jimo.2019132

[7]

Anna Canale, Francesco Pappalardo, Ciro Tarantino. Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials. Communications on Pure & Applied Analysis, 2021, 20 (1) : 405-425. doi: 10.3934/cpaa.2020274

[8]

Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442

[9]

Shengbing Deng, Tingxi Hu, Chun-Lei Tang. $ N- $Laplacian problems with critical double exponential nonlinearities. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 987-1003. doi: 10.3934/dcds.2020306

[10]

Fuensanta Andrés, Julio Muñoz, Jesús Rosado. Optimal design problems governed by the nonlocal $ p $-Laplacian equation. Mathematical Control & Related Fields, 2021, 11 (1) : 119-141. doi: 10.3934/mcrf.2020030

[11]

Raffaele Folino, Ramón G. Plaza, Marta Strani. Long time dynamics of solutions to $ p $-Laplacian diffusion problems with bistable reaction terms. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020403

[12]

Yutong Chen, Jiabao Su. Nontrivial solutions for the fractional Laplacian problems without asymptotic limits near both infinity and zero. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021007

[13]

Wenya Qi, Padmanabhan Seshaiyer, Junping Wang. A four-field mixed finite element method for Biot's consolidation problems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020127

[14]

Darko Dimitrov, Hosam Abdo. Tight independent set neighborhood union condition for fractional critical deleted graphs and ID deleted graphs. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 711-721. doi: 10.3934/dcdss.2019045

[15]

Maicon Sônego. Stable transition layers in an unbalanced bistable equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020370

[16]

Xiaoxian Tang, Jie Wang. Bistability of sequestration networks. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1337-1357. doi: 10.3934/dcdsb.2020165

[17]

Soonki Hong, Seonhee Lim. Martin boundary of brownian motion on Gromov hyperbolic metric graphs. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021014

[18]

Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077

[19]

Tian Ma, Shouhong Wang. Topological phase transition III: Solar surface eruptions and sunspots. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 501-514. doi: 10.3934/dcdsb.2020350

[20]

Russell Ricks. The unique measure of maximal entropy for a compact rank one locally CAT(0) space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 507-523. doi: 10.3934/dcds.2020266

2019 Impact Factor: 1.053

Metrics

  • PDF downloads (40)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]