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


Erzan A., Tuncer A.

LINEAR ALGEBRA AND ITS APPLICATIONS, cilt.586, ss.111-129, 2020 (SCI İndekslerine Giren Dergi) identifier identifier

  • Cilt numarası: 586
  • Basım Tarihi: 2020
  • Doi Numarası: 10.1016/j.laa.2019.10.023
  • Dergi Adı: LINEAR ALGEBRA AND ITS APPLICATIONS
  • Sayfa Sayıları: ss.111-129

Özet

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.