Signed graphs Gσ with nullity n(Gσ)−g(Gσ)−1

Abstract

Let $G^{\sigma }=(G,\sigma )$ be a signed graph of order $n\left(G^{\sigma }\right)$. Let denote the girth, rank, and nullity of $G^{\sigma }$ by $g\left(G^{\sigma }\right)$, $r\left(G^{\sigma }\right)$, and $\eta \left(G^{\sigma }\right)$, respectively. Recently, Chang and Li (2022) [6], characterized connected graphs G with nullity $n\left(G\right)-g\left(G\right)-1$. In this paper, we extend the results of Chang and Li to the setting of signed graphs with some new improvements. Furthermore, we characterize signed graphs $G^{\sigma }$ that satisfy the nullity conditions $\eta \left(G^{\sigma }\right)=n\left(G^{\sigma }\right)-g\left(G^{\sigma }\right)$ and $\eta \left(G^{\sigma }\right)=n\left(G^{\sigma }\right)-g\left(G^{\sigma }\right)-2$, providing distinct characterization from those of Q. Wu et al. (2022).