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The normalized Laplacians on both k-triangle graph and k-quadrilateral graph with their applications

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  • Huang, Jing
  • Li, Shuchao

Abstract

The k-triangle graph Tk(G) is obtained from a graph G by replacing each edge in G with k+1 parallel paths, in which one is of length 1 and each of the rest k paths is of length 2; whereas the k-quadrilateral graph Qk(G) is obtained from G by replacing each edge in G with k+1 parallel paths, in which one is of length 1 and each of the rest k paths is of length 3. In this paper, we completely determine the normalized Laplacian spectrum on Tk(G) (resp. Qk(G)) for any connected graph G, k ⩾ 2. As applications, the correlation between the degree-Kirchhoff index, the Kemeny’s constant and the number of spanning trees of Tk(G) (resp. Qk(G), the r-th iterative k-triangle graph Trk(G), the r-th iterative k-quadrilateral graph Qrk(G)) and those of G are derived. Our results extend those main results obtained in Xie et al. (2016) and Li and Hou (2017).

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:320:y:2018:i:c:p:213-225
    DOI: 10.1016/j.amc.2017.09.035
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    References listed on IDEAS

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    1. Xie, Pinchen & Zhang, Zhongzhi & Comellas, Francesc, 2016. "The normalized Laplacian spectrum of subdivisions of a graph," Applied Mathematics and Computation, Elsevier, vol. 286(C), pages 250-256.
    2. 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.
    3. van Dam, E.R. & Omidi, G.R., 2011. "Graphs whose normalized laplacian has three eigenvalues," Other publications TiSEM d3b7fa76-22b5-4a9a-8706-a, Tilburg University, School of Economics and Management.
    4. 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.
    5. 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.
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    Cited by:

    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. 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.
    3. 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.

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