Spectral analysis of certain spherically homogeneous graphs

Jonathan Breuer, Matthias Keller

Research output: Contribution to journalArticlepeer-review

18 Scopus citations

Abstract

We study operators on rooted graphs with a certain spherical homogeneity. These graphs are called path commuting and allow for a decomposition of the adjacency matrix and the Laplacian into a direct sum of Jacobi matrices which reflect the structure of the graph. Thus, the spectral properties of the adjacency matrix and the Laplacian can be analyzed by means of the elaborated theory of Jacobi matrices. For some examples which include antitrees, we derive the decomposition explicitly and present a zoo of spectral behavior induced by the geometry of the graph. In particular, these examples show that spectral types are not at all stable under rough isometries.

Original languageAmerican English
Pages (from-to)825-826
Number of pages2
JournalOperators and Matrices
Volume7
Issue number4
DOIs
StatePublished - 2013

Keywords

  • Absolutely continuous spectrum
  • Antitrees
  • Decomposition
  • Graph laplacian
  • Jacobi matrices
  • Remling's theorem
  • Rough isometry
  • Singular continuous spectrum
  • Symmetries

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