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The coefficients of the immanantal polynomial

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  • Yu, Guihai
  • Qu, Hui

Abstract

An expression of the coefficient of immanantal polynomial of an n × n matrix is present. Moreover, we give expressions of the coefficient of immanantal polynomials of combinatorial matrices (adjacency matrix, Laplacian matrix, signless Laplacian matrix). As applications, we show that the immanantal polynomials for Laplacian matrix and signless Laplacian matrix of bipartite graphs are the same. This is a generalization of the characteristic polynomial for Laplacian matrix and signless Laplacian matrix of bipartite graphs. Furthermore, we consider the relations between the characteristic polynomial and the immanantal polynomial for trees.

Suggested Citation

  • Yu, Guihai & Qu, Hui, 2018. "The coefficients of the immanantal polynomial," Applied Mathematics and Computation, Elsevier, vol. 339(C), pages 38-44.
    • Handle: RePEc:eee:apmaco:v:339:y:2018:i:c:p:38-44
    DOI: 10.1016/j.amc.2018.06.057
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    References listed on IDEAS

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    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. Wu, Tingzeng & Lai, Hong-Jian, 2018. "On the permanental sum of graphs," Applied Mathematics and Computation, Elsevier, vol. 331(C), pages 334-340.
    4. 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.
    5. Liu, Xiaogang & Wu, Tingzeng, 2017. "Computing the permanental polynomials of graphs," Applied Mathematics and Computation, Elsevier, vol. 304(C), pages 103-113.
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    Cited by:

    1. Tingzeng Wu & Huazhong Lü, 2019. "The Extremal Permanental Sum for a Quasi-Tree Graph," Complexity, Hindawi, vol. 2019, pages 1-4, May.
    2. Wu, Tingzeng & So, Wasin, 2019. "Unicyclic graphs with second largest and second smallest permanental sums," Applied Mathematics and Computation, Elsevier, vol. 351(C), pages 168-175.
    3. Wu, Tingzeng & Zhou, Tian & Lü, Huazhong, 2022. "Further results on the star degree of graphs," Applied Mathematics and Computation, Elsevier, vol. 425(C).
    4. Wu, Tingzeng & Yu, Yong & Gao, Xing, 2023. "The second immanantal polynomials of Laplacian matrices of unicyclic graphs," Applied Mathematics and Computation, Elsevier, vol. 451(C).

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