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A $\mathbb{T}$-gain graph is a triple $\Phi=(G,\mathbb{T},\varphi)$consisting of a graph $G=(V,E)$, the circle group $\mathbb{T}=\{z\in C:|z|=1\}$ and a gain function $\varphi:\overrightarrow{E}\rightarrow \mathbb{T}$such that $\varphi(e_{ij})=\varphi(e_{ji})^{-1}=\overline{\varphi(e_{ji})}$.The rank of $\mathbb{T}$-gain graph $\Phi$, denoted by $r(\Phi)$, is the rankof the adjacency matrix of $\Phi$. In 2015, Yu, Qu and Tu [ G. H. Yu, H. Qu, J.H. Tu, Inertia of complex unit gain graphs, Appl. Math. Comput. 265(2015)619--629 ] obtained some properties of inertia of a $\mathbb{T}$-gain graph.They characterized the $\mathbb{T}$-gain unicyclic graphs with small positiveor negative index. Motivated by above, in this paper, we characterize thecomplex unit gain bicyclic graphs with rank 2, 3 or 4.

Complex unit gain bicyclic graphs with rank 2, 3 or 4