Complex unit gain graphs with exactly one positive eigenvalue

Abstract A complex unit gain graph is a graph where each orientation of an edge is given a complex unit, which is the inverse of the complex unit assigned to the opposite orientation. In this paper, we characterize the structure of the complex unit gain graphs with exactly one positive eigenvalue. As its applications, we obtain the complex unit gain graphs with rank 2, and investigate the complex unit gain graphs with exactly two eigenvalues different from 0 and −1.