A complex unit gain graph (or $\mathbb{T}$-gain graph) is a triple $\mathrm{\Phi }=(G,\mathbb{T},\varphi )$ (or $(G,\varphi )$ for short) consisting of a simple graph G with $|V\left(G\right)|=n$, as the underlying graph of $(G,\varphi )$, the set of unit complex numbers $\mathbb{T}=\{z\in C:|z|=1\}$ and a gain function $\varphi :\overrightarrow{E}\to \mathbb{T}$ with the property that $\varphi \left(e_{ij}\right)=\varphi (e_{ji}{)}^{-1}=\overline{\varphi \left(e_{ji}\right)}$. The adjacency matrix of $(G,\varphi )$ is $A(G,\varphi )=(a_{ij}{)}_{n\times n}$, where $a_{ij}=\varphi \left(e_{ij}\right)$ if $v_i$ is adjacent to $v_j$ and $a_{ij}=0$ otherwise. The rank of $(G,\varphi )$, denoted by $r(G,\varphi )$, is the rank of $A(G,\varphi )$. Let $\alpha \left(G\right)$ and $c\left(G\right)$ be the independence number and the cyclomatic number of G, respectively. In this paper, we prove that $2n-2c\left(G\right)\le r(G,\varphi )+2\alpha \left(G\right)\le 2n$. And the properties of the complex unit gain graphs that attain the lower bound are characterized. Furthermore, the lower and upper bounds on $r(G,\varphi )+\alpha \left(G\right)$, $r(G,\varphi )-\alpha \left(G\right)$ and $\frac{r(G,\varphi )}{\alpha \left(G\right)}$ are identified. These results generalize the corresponding known results about undirected graphs, mixed graphs, oriented graphs and signed graphs.

复数单位增益图（或 $\mathbb{Ť}$-收益图）是三元组 $\mathrm{\Phi }=（G，\mathbb{Ť}，\varphi ）$ （要么 $（G，\varphi ）$简称）由一个简单的图形G组成，$|V（G）|=ñ$，作为的基础图 $（G，\varphi ）$，单位复数的集合 $\mathbb{Ť}=\{ž\in C：|ž|=\mathrm{1个}\}$ 和增益函数 $\varphi ：\overrightarrow{Ë}\to \mathbb{Ť}$ 具有的财产 $\varphi （{Ë}_{\mathrm{一世}Ĵ}）=\varphi （{Ë}_{Ĵ\mathrm{一世}}{）}^{-\mathrm{1个}}=\overline{\varphi （{Ë}_{Ĵ\mathrm{一世}}）}$。的邻接矩阵$（G，\varphi ）$ 是 $\mathrm{一种}（G，\varphi ）=（{\mathrm{一种}}_{\mathrm{一世}Ĵ}{）}_{ñ\times ñ}$，在哪里 ${\mathrm{一种}}_{\mathrm{一世}Ĵ}=\varphi （{Ë}_{\mathrm{一世}Ĵ}）$ 如果 $v_{\mathrm{一世}}$ 与...相邻 $v_{Ĵ}$ 和 ${\mathrm{一种}}_{\mathrm{一世}Ĵ}=0$除此以外。的等级$（G，\varphi ）$，表示为 $\mathrm{[R}（G，\varphi ）$，是 $\mathrm{一种}（G，\varphi ）$。让$\alpha （G）$ 和 $C（G）$分别是G的独立数和环数。在本文中，我们证明$2ñ-2C（G）\le \mathrm{[R}（G，\varphi ）+2\alpha （G）\le 2ñ$。并表征了达到下限的复数单位增益图的性质。此外，上下限$\mathrm{[R}（G，\varphi ）+\alpha （G）$， $\mathrm{[R}（G，\varphi ）-\alpha （G）$ 和 $\frac{\mathrm{[R}（G，\varphi ）}{\alpha （G）}$被识别。这些结果概括了有关无向图，混合图，有向图和有符号图的相应已知结果。