Relation between the H-rank of a mixed graph and the girth of its underlying graph

Abstract

Let ${\Sigma }^{\pi }=(V\left({\Sigma }^{\pi }\right),E\left({\Sigma }^{\pi }\right))$ be a mixed graph obtained from a simple graph $\Gamma$ with the same vertex set $V\left(\Gamma \right)$ and an edge set $E\left(\Gamma \right)$ containing undirected edges and arcs. Let $H_A\left({\Sigma }^{\pi }\right)$ be the (first kind of) Hermitian adjacency matrix of ${\Sigma }^{\pi }$. The $H$-rank of ${\Sigma }^{\pi }$ is the rank of $H_A\left({\Sigma }^{\pi }\right)$, denoted by $r^H\left({\Sigma }^{\pi }\right)$. The girth of $\Gamma$ is the length of the shortest cycle in $\Gamma$, dented by $g\left(\Gamma \right)$ (or simply by $g$). In this paper, we show that under some conditions the $H$-rank of a mixed graph is equal to the girth of its underlying graph. Moreover, we characterize mixed graphs with $H$-rank $g-1$ and $g+2$, distinct from the characterization of $T$-gain graphs provided by Khan (2024).