Complex unit gain graphs of rank 2

Abstract

Let T be the group of all complex numbers z with |z|=1. A complex unit gain graph, or simply a T-gain graph, is a triple Φ=(G,T,φ) consisting of a graph G=(V,E), the circle group T and a gain function φ:E→→T such that φ(vivj)=φ(vjvi)−1=φ(vjvi)‾ for any adjacent vertices vi and vj. The adjacency matrix A(Φ) of the T-gain graph Φ=(G,T,φ) of order n is an n×n complex matrix (aij), where aij=aji‾=φ(vivj) if vi is adjacent to vj and aij=0 if otherwise. The rank of a T-gain graph Φ, denoted by r(Φ), is the rank of the adjacency matrix of Φ. Yu et al. [24] obtained some properties of inertia indexes of a T-gain graph and they characterized the T-gain unicyclic graphs with small positive or negative index. Lu [8] characterized the T-gain bicyclic graphs with rank 2,3 or 4. Motivated by above, we determine the rank of a T-gain unicyclic graph and classify the T-gain graphs with rank 2.