Some Criteria for a Signed Graph to Have Full Rank

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.