Spectral analysis for a class of weighted non-p.c.f. Sierpiński-like networks

Abstract Much information about the structural properties and relevant dynamical aspects of a network can be encoded by the eigenvalues of its normalized Laplacian matrix, particularly those related to random walks. This paper presents a study of the normalized Laplacian spectra and their applications for a class of weighted non-p.c.f. Sierpiński-like networks with a non-uniformly bounded distribution of vertex degrees. Utilizing the spectral decimation technique alongside a theoretical matrix analysis supported by symbolic and numerical computations, we analytically obtain all the eigenvalues, as well as multiplicities. We then apply the derived spectra to derive the analytical expressions for Kemeny’s constant and the number of spanning trees in the networks under consideration, and calculate the asymptotic growth constant characterizing the network, demonstrating the effectiveness of our normalized Laplacian spectra computation.