The rank of a complex unit gain graph in terms of the matching number

Abstract

A complex unit gain graph (or ${\mathbb T}$-gain graph) is a triple $\Phi=(G, {\mathbb T}, \varphi)$ (or $(G, \varphi)$ for short) consisting of a simple graph $G$, as the underlying graph of $(G, \varphi)$, the set of unit complex numbers $\mathbb{T}= \{ z \in C:|z|=1 \}$ and a gain function $\varphi: \overrightarrow{E} \rightarrow \mathbb{T}$ with the property that $\varphi(e_{i,j})=\varphi(e_{j,i})^{-1}$. In this paper, we prove that $2m(G)-2c(G) \leq r(G, \varphi) \leq 2m(G)+c(G)$, where $r(G, \varphi)$, $m(G)$ and $c(G)$ are the rank of the Hermitian adjacency matrix $H(G, \varphi)$, the matching number and the cyclomatic number of $G$, respectively. Furthermore, the complex unit gain graphs $(G, \mathbb{T}, \varphi)$ with $r(G, \varphi)=2m(G)-2c(G)$ and $r(G, \varphi)=2m(G)+c(G)$ are characterized. These results generalize the corresponding known results about undirected graphs, mixed graphs and signed graphs. Moreover, we show that $2m(G-V_{0}) \leq r(G, \varphi) \leq 2m(G)+b(G)$ holds for any subset $V_0$ of $V(G)$ such that $G-V_0$ is acyclic and $b(G)$ is the minimum integer $|S|$ such that $G-S$ is bipartite for $S \subset V(G)$.