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The normalized Laplacian, degree-Kirchhoff index and spanning trees of the linear polyomino chains

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  • Huang, Jing
  • Li, Shuchao
  • Li, Xuechao

Abstract

Let Bn be a linear polyomino chain with n squares. In this paper, according to the decomposition theorem of normalized Laplacian polynomial, we obtain that the normalized Laplacian spectrum of Bn consists of the eigenvalues of two symmetric tridiagonal matrices of order n+1. Together with the relationship between the roots and coefficients of the characteristic polynomials of the above two matrices, explicit closed formulas of the degree-Kirchhoff index and the number of spanning trees of Bn are derived. Furthermore, it is interesting to find that the degree-Kirchhoff index of Bn is approximately one half of its Gutman index.

Suggested Citation

  • Huang, Jing & Li, Shuchao & Li, Xuechao, 2016. "The normalized Laplacian, degree-Kirchhoff index and spanning trees of the linear polyomino chains," Applied Mathematics and Computation, Elsevier, vol. 289(C), pages 324-334.
    • Handle: RePEc:eee:apmaco:v:289:y:2016:i:c:p:324-334
    DOI: 10.1016/j.amc.2016.05.024
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    References listed on IDEAS

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    1. Xie, Pinchen & Zhang, Zhongzhi & Comellas, Francesc, 2016. "On the spectrum of the normalized Laplacian of iterated triangulations of graphs," Applied Mathematics and Computation, Elsevier, vol. 273(C), pages 1123-1129.
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    Cited by:

    1. Liu, Jia-Bao & Zhao, Jing & Cai, Zheng-Qun, 2020. "On the generalized adjacency, Laplacian and signless Laplacian spectra of the weighted edge corona networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 540(C).
    2. Li, Danyi & Yan, Weigen, 2023. "Counting spanning trees with a Kekulé structure in linear hexagonal chains," Applied Mathematics and Computation, Elsevier, vol. 456(C).
    3. He, Weihua & Li, Hao & Xiao, Shuofa, 2017. "On the minimum Kirchhoff index of graphs with a given vertex k-partiteness and edge k-partiteness," Applied Mathematics and Computation, Elsevier, vol. 315(C), pages 313-318.
    4. Yasir Ahmad & Umar Ali & Daniele Ettore Otera & Xiang-Feng Pan, 2024. "Study of Random Walk Invariants for Spiro-Ring Network Based on Laplacian Matrices," Mathematics, MDPI, vol. 12(9), pages 1-19, April.
    5. Ma, Xiaoling & Bian, Hong, 2019. "The normalized Laplacians, degree-Kirchhoff index and the spanning trees of hexagonal Möbius graphs," Applied Mathematics and Computation, Elsevier, vol. 355(C), pages 33-46.
    6. Jia-Bao Liu & Jing Zhao & Zhongxun Zhu & Jinde Cao, 2019. "On the Normalized Laplacian and the Number of Spanning Trees of Linear Heptagonal Networks," Mathematics, MDPI, vol. 7(4), pages 1-15, March.
    7. Li, Zhemin & Xie, Zheng & Li, Jianping & Pan, Yingui, 2020. "Resistance distance-based graph invariants and spanning trees of graphs derived from the strong prism of a star," Applied Mathematics and Computation, Elsevier, vol. 382(C).
    8. Huang, Jing & Li, Shuchao, 2018. "The normalized Laplacians on both k-triangle graph and k-quadrilateral graph with their applications," Applied Mathematics and Computation, Elsevier, vol. 320(C), pages 213-225.
    9. Li, Deqiong & Hou, Yaoping, 2017. "The normalized Laplacian spectrum of quadrilateral graphs and its applications," Applied Mathematics and Computation, Elsevier, vol. 297(C), pages 180-188.
    10. Huang, Guixian & He, Weihua & Tan, Yuanyao, 2019. "Theoretical and computational methods to minimize Kirchhoff index of graphs with a given edge k-partiteness," Applied Mathematics and Computation, Elsevier, vol. 341(C), pages 348-357.
    11. Hong, Yunchao & Zhu, Zhongxun & Luo, Amu, 2018. "Some transformations on multiplicative eccentricity resistance-distance and their applications," Applied Mathematics and Computation, Elsevier, vol. 323(C), pages 75-85.
    12. Fei, Junqi & Tu, Jianhua, 2018. "Complete characterization of bicyclic graphs with the maximum and second-maximum degree Kirchhoff index," Applied Mathematics and Computation, Elsevier, vol. 330(C), pages 118-124.

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