IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v7y2019i4p314-d217931.html
   My bibliography  Save this article

On the Normalized Laplacian and the Number of Spanning Trees of Linear Heptagonal Networks

Author

Listed:
  • Jia-Bao Liu

    (School of Mathematical Sciences, Anhui Jianzhu University, Hefei 230601, China
    School of Mathematics, Southeast University, Nanjing 210096, China)

  • Jing Zhao

    (School of Mathematical Sciences, Anhui Jianzhu University, Hefei 230601, China)

  • Zhongxun Zhu

    (Department of Mathematics and Statistics, South Central University for Nationalities, Wuhan 430074, China)

  • Jinde Cao

    (School of Mathematics, Southeast University, Nanjing 210096, China)

Abstract

The normalized Laplacian plays an important role on studying the structure properties of non-regular networks. In fact, it focuses on the interplay between the structure properties and the eigenvalues of networks. Let H n be the linear heptagonal networks. It is interesting to deduce the degree-Kirchhoff index and the number of spanning trees of H n due to its complicated structures. In this article, we aimed to first determine the normalized Laplacian spectrum of H n by decomposition theorem and elementary operations which were not stated in previous results. We then derived the explicit formulas for degree-Kirchhoff index and the number of spanning trees with respect to H n .

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:4:p:314-:d:217931
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/7/4/314/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/7/4/314/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. 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.
    2. 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.
    3. 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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Sun, Shaowei & Das, Kinkar Ch., 2019. "On the second largest normalized Laplacian eigenvalue of graphs," Applied Mathematics and Computation, Elsevier, vol. 348(C), pages 531-541.
    2. 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.
    3. 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).
    4. 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.
    5. Sardar, Muhammad Shoaib & Pan, Xiang-Feng & Xu, Si-Ao, 2020. "Computation of resistance distance and Kirchhoff index of the two classes of silicate networks," Applied Mathematics and Computation, Elsevier, vol. 381(C).
    6. 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.
    7. Wang, Chengyong & Guo, Ziliang & Li, Shuchao, 2018. "Expected hitting times for random walks on the k-triangle graph and their applications," Applied Mathematics and Computation, Elsevier, vol. 338(C), pages 698-710.
    8. 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).
    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. 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.
    12. Li, Danyi & Yan, Weigen, 2023. "Counting spanning trees with a Kekulé structure in linear hexagonal chains," Applied Mathematics and Computation, Elsevier, vol. 456(C).
    13. 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.
    14. 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.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:7:y:2019:i:4:p:314-:d:217931. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.