Eur. Phys. J. B 63, 507-513 (2008)
https://doi.org/10.1140/epjb/e2008-00262-2

Topologies and Laplacian spectra of a deterministic uniform recursive tree

¹ Department of Computer Science and Engineering, Fudan University, Shanghai, 200433, P.R. China
² Shanghai Key Lab of Intelligent Information Processing, Fudan University, Shanghai, 200433, P.R. China
³ Department of Computer Science and Technology, Tongji University, 4800 Cao'an Road, Shanghai, 201804, P.R. China

Corresponding authors: ^a zhangzz@fudan.edu.cn - ^b sgzhou@fudan.edu.cn

Received: 27 January 2008
Revised: 6 May 2008
Published online: 10 July 2008

Abstract

The uniform recursive tree (URT) is one of the most important models and has been successfully applied to many fields. Here we study exactly the topological characteristics and spectral properties of the Laplacian matrix of a deterministic uniform recursive tree, which is a deterministic version of URT. Firstly, from the perspective of complex networks, we determine the main structural characteristics of the deterministic tree. The obtained vigorous results show that the network has an exponential degree distribution, small average path length, power-law distribution of node betweenness, and positive degree-degree correlations. Then we determine the complete Laplacian spectra (eigenvalues) and their corresponding eigenvectors of the considered graph. Interestingly, all the Laplacian eigenvalues are distinct.