Inertias of Laplacian matrices of weighted signed graphs

K Hassani Monfared,G Macgillivray,P Van Den Driessche,D D Olesky

doi:10.1515/spma-2019-0026

Abstract

Abstract We study the sets of inertias achieved by Laplacian matrices of weighted signed graphs. First we characterize signed graphs with a unique Laplacian inertia. Then we show that there is a sufficiently small perturbation of the nonzero weights on the edges of any connected weighted signed graph so that all eigenvalues of its Laplacian matrix are simple. Next, we give upper bounds on the number of possible Laplacian inertias for signed graphs with a fixed flexibility τ (a combinatorial parameter of signed graphs), and show that these bounds are sharp for an infinite family of signed graphs. Finally, we provide upper bounds for the number of possible Laplacian inertias of signed graphs in terms of the number of vertices.

Highlights

We study the sets of inertias achieved by Laplacian matrices of weighted signed graphs
The signs of the real parts of the eigenvalues of a coe cient matrix in a system of linear ordinary di erential equations determine the stability of the dynamical system that it is describing
We focus on systems with a coe cient matrix that is a symmetric Laplacian matrix, for which all eigenvalues are real

Summary

Introduction

The signs of the real parts of the eigenvalues of a coe cient matrix in a system of linear ordinary di erential equations determine the stability of the dynamical system that it is describing. When G has only positive edges, L(Γ) is a symmetric negative semi-de nite matrix where the multiplicity of the zero eigenvalue (nullity) is equal to the number of connected components of G. The Laplacian inertia of a weighted signed graph Γ = (G, γ) on n vertices is IL(Γ) = (n+, n−, n ), where n+ is the number of positive eigenvalues of L(Γ), n− is the number of negative eigenvalues of L(Γ), and n = n − n+ − n− ≥ is the multiplicity of 0 as an eigenvalue of L(Γ). For any consistent weighting γ, the following inequalities hold for the components of the Laplacian inertia (n+, n−, n ) of the weighted signed graph Γ = (G, γ): c+ − c ≤ n+ ≤ n − c−. Observe that the weighting of Γ(t) is a consistent weighting of G if only if t >

De ne the crossing polynomial

Signed Graphs with Unique Laplacian Inertias

Upper bounds for the number of Laplacian Inertias