Bounds for the rank of a complex unit gain graph in terms of the independence number

A complex unit gain graph (or $mathbb{T}$-gain graph) is a triple $Phi=(G, mathbb{T}, varphi)$ ($(G, varphi)$ for short) consisting of a graph $G$ as the underlying graph of $(G, varphi)$, $mathbb{T}= { z in C:|z|=1 } $ is a subgroup of the multiplicative group of all nonzero complex numbers $mathbb{C}^{times}$ and a gain function $varphi: overrightarrow{E} rightarrow mathbb{T}$ such that $varphi(e_{ij})=varphi(e_{ji})^{-1}=overline{varphi(e_{ji})}$. In this paper, we investigate the relation among the rank, the independence number and the cyclomatic number of a complex unit gain graph $(G, varphi)$ with order $n$, and prove that $2n-2c(G) leq r(G, varphi)+2alpha(G) leq 2n$. Where $r(G, varphi)$, $alpha(G)$ and $c(G)$ are the rank of the Hermitian adjacency matrix $A(G, varphi)$, the independence number and the cyclomatic number of $G$, respectively. Furthermore, the properties of the complex unit gain graph that reaching the lower bound are characterized.