Recursive solutions for Laplacian spectra and eigenvectors of a class of growing treelike networks

Abstract

The complete knowledge of Laplacian eigenvalues and eigenvectors of complex networks plays an outstanding role in understanding various dynamical processes running on them; however, determining analytically Laplacian eigenvalues and eigenvectors is a theoretical challenge. In this paper, we study the Laplacian spectra and their corresponding eigenvectors of a class of deterministically growing treelike networks. The two interesting quantities are determined through the recurrence relations derived from the structure of the networks. Beginning from the rigorous relations one can obtain the complete eigenvalues and eigenvectors for the networks of arbitrary size. The analytical method opens the way to analytically compute the eigenvalues and eigenvectors of some other deterministic networks, making it possible to accurately calculate their spectral characteristics.

Definitive version accepted for publication in Physical Reivew E

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• Measuring the dimension of partially embedded networks
• Exact evaluation of the causal spectrum and localization properties of electronic states on a scale-free network
• Structural properties and average tapping time on scale-free graphs with smallest diameter