Singularity of Hermitian (quasi-)Laplacian matrix of mixed graphs

Guihai Yu, Xin Liu and Hui Qu

Applied Mathematics and Computation, 2017, vol. 293, issue C, 287-292

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 an 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. Let D(M)=diag(d1,d2,…,dn) be a diagonal matrix where di is the degree of vertex vi in the underlying graph Mu. Hermitian matrices L(M)=D(M)−H(M),Q(M)=D(M)+H(M) are said as the Hermitian Laplacian matrix, Hermitian quasi-Laplacian matrix of mixed graph M, respectively. In this paper, it is shown that they are positive semi-definite. Moreover, we characterize the singularity of them. In addition, an expression of the determinant of the Hermitian (quasi-)Laplacian matrix is obtained.

Keywords: Mixed graphs; Hermitian adjacency matrix; Hermitian (quasi-)Laplacian matrix (search for similar items in EconPapers)
Date: 2017
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (5)

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0096300316305276
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:293:y:2017:i:c:p:287-292

DOI: 10.1016/j.amc.2016.08.032

Access Statistics for this article

Applied Mathematics and Computation is currently edited by Theodore Simos

More articles in Applied Mathematics and Computation from Elsevier
Bibliographic data for series maintained by Catherine Liu ().