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 ) = ( h k l ) , where h k l = - h l k = i ( i = - 1 ) if there exists an orientation from vk to vl and h k l = h l k = 1 if there exists an edge between vk and vl but not exist any orientation, and h k l = 0 otherwise. The value of a mixed walk W = v 1 v 2 v 3 ? v l is h ( W ) = h 12 h 23 ? 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.

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