On Spectral Properties of Signed Laplacians with Connections to Eventual Positivity

TL;DR

This paper explores the spectral characteristics of signed Laplacians in undirected graphs, establishing conditions for positive semidefiniteness, characterizing inertia for indefinite cases, and linking these matrices to generalized M-matrices and exponential positivity.

Contribution

It introduces a comprehensive framework connecting signed Laplacians with network theory, M-matrices, and positivity, providing new insights into their spectral properties.

Findings

Conditions for positive semidefiniteness of signed Laplacians

Characterization of inertia for indefinite signed Laplacians

Connections between signed Laplacians, M-matrices, and exponential positivity

Abstract

Signed graphs have appeared in a broad variety of applications, ranging from social networks to biological networks, from distributed control and computation to power systems. In this paper, we investigate spectral properties of signed Laplacians for undirected signed graphs. We find conditions on the negative weights under which a signed Laplacian is positive semidefinite via the Kron reduction and multiport network theory. For signed Laplacians that are indefinite, we characterize their inertias with the same framework. Furthermore, we build connections between signed Laplacians, generalized M-matrices, and eventually exponentially positive matrices.