Inertia of complex unit gain graphs

摘要

Let T={z∈C:|z|=1} be a subgroup of the multiplicative group of all nonzero complex numbers C×. 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(eij)=φ(eji)−1=φ(eji)¯. The adjacency matrix A(Φ) of the T-gain graph Φ=(G,φ) of order n is an n × n complex matrix (aij), where aij={φ(eij),ifviisadjacenttovj,0,otherwise.Evidently this matrix is Hermitian. The inertia of Φ is defined to be the triple In(Φ)=(i+(Φ),i−(Φ),i0(Φ)), where i+(Φ),i−(Φ),i0(Φ) are numbers of the positive, negative and zero eigenvalues of A(Φ) including multiplicities, respectively. In this paper we investigate some properties of inertia of T-gain graph.