Hermitian Laplacian matrix and positive of mixed graphs

Guihai Yu and Hui Qu

Applied Mathematics and Computation, 2015, vol. 269, issue C, 70-76

Abstract: 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.

Keywords: Mixed graph; Hermitian adjacency matrix; Hermitian Laplacian matrix; Positive of mixed graph (search for similar items in EconPapers)
Date: 2015
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (3)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:269:y:2015:i:c:p:70-76

DOI: 10.1016/j.amc.2015.07.045

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