A moment inequality and positivity for signed graph Laplacians

TL;DR

This paper establishes an eigenvalue inequality for signed graph Laplacians based on edge weight moments, providing conditions for positive semi-definiteness and insights into spectral properties of various graph models.

Contribution

It introduces a new eigenvalue inequality for signed graph Laplacians involving moments of edge weights and spectral characteristics, extending classical results.

Findings

Inequality bounds eigenvalues using edge weight moments

Results are tight for Erdős-Rényi and regular graphs

Provides spectral insights for random and complete graphs

Abstract

A number of recent papers have considered signed graph Laplacians, a generalization of the classical graph Laplacian, where the edge weights are allowed to take either sign. In the classical case, where the edge weights are all positive, the Laplacian is positive semi-definite with the dimension of the kernel representing the number of connected components of the graph. In many applications one is interested in establishing conditions which guarantee the positive semi-definiteness of the matrix. In this paper we present an inequality on the eigenvalues of a weighted graph Laplacian (where the weights need not have any particular sign) in terms of the first two moments of the edge weights. This bound involves the eigenvalues of the equally weighted Laplacian on the graph as well as the eigenvalues of the adjacency matrix of the line graph (the edge-to-vertex dual graph). For a regular graph the bound can be expressed entirely in terms of the second eigenvalue of the equally weighted Laplacian, an object that has been extensively studied in connection with expander graphs and spectral measures of graph connectivity. We present several examples including Erd\H{o}s-R\'enyi random graphs in the critical and subcritical regimes, random large $d$-regular graphs, and the complete graph, for which the inequalities here are tight.