Hermitian Laplacian matrix and positive of mixed graphs

A mixed graph is obtained from an undirected graph by orienting a subset of its edges. The Hermitian adjacency matrix of a mixed graph M of order n is the n × n matrix H(M)=(hkl), where hkl=−hlk=i (i=−1) if there exists an orientation from vk to vl and hkl=hlk=1 if there exists an edge between vk and vl but not exist any orientation, and hkl=0 otherwise. The value of a mixed walk W=v1v2v3⋯vl is h(W)=h12h23⋯h(l−1)l. A mixed walk is positive (negative) if h(W)=1 (h(W)=−1). A mixed cycle is called positive if its value is 1. A mixed graph is positive if each of its mixed cycle is positive. In this work we firstly present the necessary and sufficient conditions for the positive of a mixed graph. Secondly we introduce the incident matrix and Hermitian Laplacian matrix of a mixed graph and derive some results about the Hermitian Laplacian spectrum.