The Laplacian Spectra of Graphs and Complex Networks

TL;DR

This survey reviews recent advances in understanding the Laplacian spectra of graphs and complex networks, including random graphs and small world networks, highlighting key spectral properties and open questions.

Contribution

It summarizes recent results on Laplacian spectra, including spectral radius, coefficients, algebraic connectivity, and spectra of specific network models, providing a comprehensive overview.

Findings

Spectral radius bounds for given degree sequences

Analysis of Laplacian coefficients and algebraic connectivity

Spectral properties of random and small world networks

Abstract

The paper is a brief survey of some recent new results and progress of the Laplacian spectra of graphs and complex networks (in particular, random graph and the small world network). The main contents contain the spectral radius of the graph Laplacian for given a degree sequence, the Laplacian coefficients, the algebraic connectivity and the graph doubly stochastic matrix, and the spectra of random graphs and the small world networks. In addition, some questions are proposed.