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Hermitian Laplacian matrix and positive of mixed graphs

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  • Yu, Guihai
  • 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 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.

Suggested Citation

  • Yu, Guihai & Qu, Hui, 2015. "Hermitian Laplacian matrix and positive of mixed graphs," Applied Mathematics and Computation, Elsevier, vol. 269(C), pages 70-76.
    • Handle: RePEc:eee:apmaco:v:269:y:2015:i:c:p:70-76
    DOI: 10.1016/j.amc.2015.07.045
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    References listed on IDEAS

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    1. Chen, Zengqiang & Dehmer, Matthias & Shi, Yongtang, 2015. "Bounds for degree-based network entropies," Applied Mathematics and Computation, Elsevier, vol. 265(C), pages 983-993.
    2. Tsang, Eric W. K., 2014. "Old and New," Management and Organization Review, Cambridge University Press, vol. 10(03), pages 390-390, November.
    3. Shi, Yongtang, 2015. "Note on two generalizations of the Randić index," Applied Mathematics and Computation, Elsevier, vol. 265(C), pages 1019-1025.
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    1. 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.
    2. 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.
    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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