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

A T-gain graph is a triple Φ=(G,T,φ) consisting of a graph G=(V,E), the circle group T={z∈C:|z|=1} and a gain function φ:E→→T such that φ(eij)=φ(eji)−1=φ(eji)‾. The rank of T-gain graph Φ, denoted by r(Φ), is the rank of the adjacency matrix of Φ. Yu et al. (2015) [8] obtained some properties of inertia of a T-gain graph. They characterized the T-gain unicyclic graphs with small positive or negative index. Motivated by above, in this paper, we characterize the complex unit gain connected bicyclic graphs with rank 2, 3 or 4.