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 (SCI-Expanded) identifier identifier identifier

  • Publication Type: Article / Article
  • Volume: 586
  • Publication Date: 2020
  • Doi Number: 10.1016/j.laa.2019.10.023
  • Journal Name: LINEAR ALGEBRA AND ITS APPLICATIONS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Applied Science & Technology Source, Compendex, Computer & Applied Sciences, MathSciNet, zbMATH
  • Page Numbers: pp.111-129
  • Istanbul Technical University Affiliated: Yes

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.