  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. Such analytical solutions for
the eigenvectors of non-periodic networks are needed to provide a firm basis
for the spectral renormalization group which we have proposed earlier [A.
Tuncer and A. Erzan, Phys. Rev. E {\bf 92}, 022106 (2015)].
  PACS Nos. 02.10.Ox Combinatorics; graph theory, 02.10.Ud Linear algebra,
02.30 Nw Fourier analysis
