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

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 $S subset V(G)$.