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

A $\mathbb{T}$-gain graph is a triple $Φ=(G,\mathbb{T},φ)$ consisting of a graph $G=(V,E)$, the circle group $\mathbb{T}=\{z\in C: |z|=1\}$ and a gain function $φ:\overrightarrow{E}\rightarrow \mathbb{T}$ such that $φ(e_{ij})=φ(e_{ji})^{-1}=\overline{φ(e_{ji})}$. The rank of $\mathbb{T}$-gain graph $Φ$, denoted by $r(Φ)$, is the rank of the adjacency matrix of $Φ$. 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.

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