Inertia of complex unit gain graphs

Let T={z∈C:|z|=1}T={z∈C:|z|=1} be a subgroup of the multiplicative group of all nonzero complex numbers C×C×. A TT-gain graph is a triple Φ=(G,T,φ)Φ=(G,T,φ) consisting of a graph G=(V,E),G=(V,E), the circle group TT and a gain function φ:E→→T such that φ(eij)=φ(eji)−1=φ(eji)¯. The adjacency matrix A(Φ  ) of the TT-gain graph Φ=(G,φ)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)),In(Φ)=(i+(Φ),i−(Φ),i0(Φ)), where i+(Φ),i−(Φ),i0(Φ)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 TT-gain graph.