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 $(G,\phi )$, the set of unit complex numbers $\mathbb{T}=\{z\in C:\mathrm{\mid }z\mathrm{\mid }=1\}$ and a gain function $\varphi: \overrightarrow{E} \rightarrow $\mathbb{T}$$withthepropertythat$\varphi(e_{i,j})=\varphi(e_{j,i})^{-1}$\mathrm{.}Inthispaper,weprovethat$2m(G)-2c(G) \leq r(G, \varphi) \leq 2m(G)+c(G)$,where$r(G, \varphi)$,$m(G)$and$c(G)$aretherankoftheHermitianadjacencymatrix$H(G, \varphi)$, the matching number and the cyclomatic number of $G$, respectively. Furthermore, the complex unit gain graphs $(G,\mathbb{T},\phi )$ 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 $2m(G-V_{0}) \leq r(G, \varphi) \leq 2m(G)+b(G)$holdsforanysubset$V_0$of$V(G)$suchthat$G-V_0$isacyclicand$b(G)$istheminimuminteger$|S|$suchthat$G-S$ is bipartite for $S\subset V(G)$.