Some Criteria for a Signed Graph to Have Full Rank

TL;DR

This paper characterizes when signed and weighted graphs have adjacency matrices of full rank, linking these properties to the existence and number of certain factors within the underlying graph.

Contribution

It provides necessary and sufficient conditions for graphs to have signings or weightings that produce full or non-full rank adjacency matrices, based on factor properties.

Findings

A graph has a signing with full rank adjacency matrix iff it has a {1,2}-factor.

A graph admits a weighting with non-full rank adjacency matrix iff it has at least two {1,2}-factors.

Abstract

A weighted graph $G^{\omega}$ consists of a simple graph $G$ with a weight $\omega$, which is a mapping,$\omega$: $E(G)\rightarrow\mathbb{Z}\backslash\{0\}$. A signed graph is a graph whose edges are labeled with $-1$ or $1$. In this paper, we characterize graphs which have a sign such that their signed adjacency matrix has full rank, and graphs which have a weight such that their weighted adjacency matrix does not have full rank. We show that for any arbitrary simple graph $G$, there is a sign $\sigma$ so that $G^{\sigma}$ has full rank if and only if $G$ has a $\{1,2\}$-factor. We also show that for a graph $G$, there is a weight $\omega$ so that $G^{\omega}$ does not have full rank if and only if $G$ has at least two $\{1,2\}$-factors.