Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6420079 | Applied Mathematics and Computation | 2015 | 7 Pages |
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 à n matrix H(M)=(hkl), where hkl=âhlk=i (i=â1) 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=v1v2v3â¯vl is h(W)=h12h23â¯h(lâ1)l. A mixed walk is positive (negative) if h(W)=1 (h(W)=â1). 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.