Explicit construction of the eigenvectors and eigenvalues of the graph Laplacian on the Cayley tree


Erzan A., Tuncer A.

LINEAR ALGEBRA AND ITS APPLICATIONS, vol.586, pp.111-129, 2020 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 586
  • Publication Date: 2020
  • Doi Number: 10.1016/j.laa.2019.10.023
  • Title of Journal : LINEAR ALGEBRA AND ITS APPLICATIONS
  • Page Numbers: pp.111-129

Abstract

A generalized Fourier analysis on arbitrary graphs calls for a detailed knowledge of the eigenvectors of the graph Laplacian. Using the symmetries of the Cayley tree, we recursively construct the family of eigenvectors with exponentially growing eigenspaces, associated with eigenvalues in the lower part of the spectrum. The spectral gap decays exponentially with the tree size, for large trees. The eigenvalues and eigenvectors obey recursion relations which arise from the nested geometry of the tree. (C) 2019 Elsevier Inc. All rights reserved.