Topologies and Laplacian spectra of a deterministic uniform recursive tree

TL;DR

This paper analyzes the topological features and Laplacian spectral properties of a deterministic uniform recursive tree, revealing its structural characteristics and unique eigenvalue spectrum.

Contribution

It provides an exact analysis of the topological structure and Laplacian spectra of a deterministic version of the uniform recursive tree, which was previously unexplored.

Findings

The network exhibits exponential degree distribution.

The network has small average path length.

All Laplacian eigenvalues are distinct.

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.