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

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  • Yu, Guihai
  • Liu, Xin
  • Qu, Hui

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.

Suggested Citation

  • Yu, Guihai & Liu, Xin & Qu, Hui, 2017. "Singularity of Hermitian (quasi-)Laplacian matrix of mixed graphs," Applied Mathematics and Computation, Elsevier, vol. 293(C), pages 287-292.
    • Handle: RePEc:eee:apmaco:v:293:y:2017:i:c:p:287-292
    DOI: 10.1016/j.amc.2016.08.032
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    References listed on IDEAS

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    1. van Dam, E.R. & Haemers, W.H., 2002. "Which Graphs are Determined by their Spectrum?," Discussion Paper 2002-66, Tilburg University, Center for Economic Research.
    2. Yu, Guihai & Qu, Hui, 2015. "Hermitian Laplacian matrix and positive of mixed graphs," Applied Mathematics and Computation, Elsevier, vol. 269(C), pages 70-76.
    3. Lang, Rongling & Li, Tao & Mo, Desen & Shi, Yongtang, 2016. "A novel method for analyzing inverse problem of topological indices of graphs using competitive agglomeration," Applied Mathematics and Computation, Elsevier, vol. 291(C), pages 115-121.
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    Cited by:

    1. Cao, Shujuan & Dehmer, Matthias & Kang, Zhe, 2017. "Network Entropies Based on Independent Sets and Matchings," Applied Mathematics and Computation, Elsevier, vol. 307(C), pages 265-270.
    2. Lei, Hui & Li, Tao & Ma, Yuede & Wang, Hua, 2018. "Analyzing lattice networks through substructures," Applied Mathematics and Computation, Elsevier, vol. 329(C), pages 297-314.
    3. Lan, Yongxin & Li, Tao & Ma, Yuede & Shi, Yongtang & Wang, Hua, 2018. "Vertex-based and edge-based centroids of graphs," Applied Mathematics and Computation, Elsevier, vol. 331(C), pages 445-456.
    4. Yu, Guihai & Qu, Hui & Dehmer, Matthias, 2017. "Principal minor version of Matrix-Tree theorem for mixed graphs," Applied Mathematics and Computation, Elsevier, vol. 309(C), pages 27-30.
    5. Guihai Yu & Hui Qu, 2018. "More on Spectral Analysis of Signed Networks," Complexity, Hindawi, vol. 2018, pages 1-6, October.

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