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

A $\mathbb{T}$-gain graph is a triple $\mathrm{\Phi }=(G,\mathbb{T},\phi )$ consisting of a graph $G=(V,E)$, the circle group $$\mathbb{T}$={z\in C: |z|=1}$andagainfunction$\varphi:\overrightarrow{E}\rightarrow $\mathbb{T}$$suchthat$\varphi(e_{ij})=\varphi(e_{ji})^{-1}=\overline{\varphi(e_{ji})}$. The rank of $\mathbb{T}$-gain graph $\mathrm{\Phi }$, denoted by $r(\mathrm{\Phi })$, is the rank of the adjacency matrix of $\mathrm{\Phi }$. 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 positive or negative index. Motivated by above, in this paper, we characterize the complex unit gain bicyclic graphs with rank 2, 3 or 4.