The rank of a signed graph in terms of girth

Abstract

Let $\Gamma=(G,\sigma)$ be a signed graph and $A(G,\sigma)$ be its adjacency matrix. Denote by $gr(G)$ the girth of $G$, which is the length of the shortest cycle in $G$. Let $r(G,\sigma)$ be the rank of $(G,\sigma)$. In this paper, we will prove that $r(G,\sigma)\geq gr(G)-2$ for a signed graph $(G,\sigma)$. Moreover, we characterize all extremal graphs which satisfy the equalities $r(G,\sigma)=gr(G)-2$ and $r(G,\sigma)=gr(G)$.