On Eigenvalues of Laplacian Matrix for a Class of Directed Signed Graphs

TL;DR

This paper investigates the eigenvalues of Laplacian matrices in directed signed graphs, establishing conditions under which eigenvalues have positive real parts or become negative, with implications for graph stability.

Contribution

It introduces necessary and sufficient conditions for the eigenvalues of Laplacian matrices in directed signed graphs, especially under negative weight perturbations.

Findings

Necessary condition for positive real parts of eigenvalues.

Sufficient condition for unperturbed graphs.

Negative eigenvalues arise with infinitesimal negative weights.

Abstract

The eigenvalues of the Laplacian matrix for a class of directed graphs with both positive and negative weights are studied. First, a class of directed signed graphs is investigated in which one pair of nodes (either connected or not) is perturbed with negative weights. A necessary condition is proposed to attain the following objective for the perturbed graph: the real parts of the non-zero eigenvalues of its Laplacian matrix are positive. A sufficient condition is also presented that ensures the aforementioned objective for the unperturbed graph. It is then highlighted the case where the condition becomes necessary and sufficient. Secondly, for directed graphs, a subset of pairs of nodes are identified where if any of the pairs is connected by an edge with infinitesimal negative weight, the resulting Laplacian matrix will have at least one eigenvalue with negative real part. Illustrative examples are presented to show the applicability of our results.