On the sign patterns of the smallest signless Laplacian eigenvector

TL;DR

This paper investigates the stability of the sign pattern of the smallest eigenvector of the signless Laplacian in bipartite graphs when edges are added within the bipartition sets.

Contribution

It characterizes conditions under which the sign pattern of the eigenvector remains unchanged despite modifications to the graph.

Findings

Sign pattern persists under certain edge additions within bipartition sets.

Conditions identified for the stability of the eigenvector's sign pattern.

Provides cases where the sign pattern of the eigenvector is resilient to graph modifications.

Abstract

Let $H$ be a connected bipartite graph, whose signless Laplacian matrix is $Q(H)$. Suppose that the bipartition of $H$ is $(S,T)$ and that $x$ is the eigenvector of the smallest eigenvalue of $Q(H)$. It is well-known that $x$ is positive and constant on $S$, and negative and constant on $T$. The resilience of the sign pattern of $x$ under addition of edges into the subgraph induced by either $S$ or $T$ is investigated and a number of cases in which the sign pattern of $x$ persists are described.