2018/06/04 by Ayşe Erzan, Aslı Tuncer · 30 citations
Materials Science · Mathematics · Physics and Astronomy · #Combinatorics #Complex Network Analysis Techniques #Discrete mathematics #Eigenvalues and eigenvectors #Graph #Graph theory and applications #Laplace operator #Laplacian matrix #Magnetism in coordination complexes #Mathematical analysis #Mathematics #Tree (set theory) #cond-mat.stat-mech
paper · pdf · doi:10.1016/j.laa.2019.10.023
published in Linear Algebra and its Applications 586, 111-129 (Elsevier BV)
arxiv created 2018/06/04 · openalex publication_date 2019/10/25 · arxiv updated 2019/10/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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