Isospectral infinite graphs and network and infinite eigenvalue multiplicities

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∞-setting. Moreover, the occurrence of eigenvalue multiplicities with eigenspaces containing subspaces isomorphic to ℓ∞(Z) is investigated, in particular in trees and periodic graphs. © American Institute of Mathematical Sciences.