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Expected hitting times for random walks on the k-triangle graph and their applications

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  • Wang, Chengyong
  • Guo, Ziliang
  • Li, Shuchao

Abstract

Given a simple connected graph G, the k-triangle graph of G, written by Tk(G), is obtained from G by adding k new vertices ui1,ui2,…,uik for each edge ei=uv in G and then adding in edges uui1,uui2,…,uuik and ui1v,ui2v,…,uikv. In this paper, the eigenvalues and eigenvectors of the probability transition matrix of random walks on Tk(G) are completely determined. Then the expected hitting times between any two vertices of Tk(G) are given in terms of those of G. Using these results all the relationship on the number of spanning trees (resp. Kemeny’s constant, the degree-Kirchhoff index) in Tk(G) compared to those of G is found. As well the resistance distance between any two vertices of Tk(G) is given with respect to those of G.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:338:y:2018:i:c:p:698-710
    DOI: 10.1016/j.amc.2018.06.056
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    References listed on IDEAS

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    1. Palacios, JoséLuis & Tetali, Prasad, 1996. "A note on expected hitting times for birth and death chains," Statistics & Probability Letters, Elsevier, vol. 30(2), pages 119-125, October.
    2. Miguel Río & José Luis Palacios, 2016. "Decomposing Hitting Times of Walks on Graphs into Simpler Ones," Methodology and Computing in Applied Probability, Springer, vol. 18(4), pages 1035-1042, December.
    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.
    4. 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.
    5. 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. Lv, Yan & Chen, Zhouyang & Wu, Tingzeng & Zhang, Peng-Li, 2024. "On the weighted reverse cover cost of trees and unicyclic graphs with given diameter," Applied Mathematics and Computation, Elsevier, vol. 473(C).

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