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Theoretical and computational methods to minimize Kirchhoff index of graphs with a given edge k-partiteness

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  • Huang, Guixian
  • He, Weihua
  • Tan, Yuanyao

Abstract

Let G be a connected graph. The edge k-partiteness of G is the minimum number of edges whose deletion from G yields a k-partite graph. The Kirchhoff index Kf(G) of G is the sum of the resistance distance between all unordered pairs of vertices. In this paper, we study the problem to determine the minimum Kirchhoff index of graphs with a given edge k-partiteness. First, we theoretically characterize the graphs with minimum Kirchhoff index in this graph family when the edge k-partiteness is not big compared to the number of vertices of G. Then we propose an exhaustive search algorithm to find the optimal graphs. At last, three strategies are used to reduce the computation complexity of our algorithm and several performance comparisons of these strategies are given.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:341:y:2019:i:c:p:348-357
    DOI: 10.1016/j.amc.2018.09.002
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    References listed on IDEAS

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    1. Qi, Xuli & Zhou, Bo & Du, Zhibin, 2016. "The Kirchhoff indices and the matching numbers of unicyclic graphs," Applied Mathematics and Computation, Elsevier, vol. 289(C), pages 464-480.
    2. Liu, Jia-Bao & Pan, Xiang-Feng, 2016. "Minimizing Kirchhoff index among graphs with a given vertex bipartiteness," Applied Mathematics and Computation, Elsevier, vol. 291(C), pages 84-88.
    3. Yang, Yujun & Cao, Yuliang & Yao, Haiyuan & Li, Jing, 2018. "Solution to a conjecture on a Nordhaus–Gaddum type result for the Kirchhoff index," Applied Mathematics and Computation, Elsevier, vol. 332(C), pages 241-249.
    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. José Luis Palacios, 2004. "Foster's Formulas via Probability and the Kirchhoff Index," Methodology and Computing in Applied Probability, Springer, vol. 6(4), pages 381-387, December.
    6. 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.
    7. 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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    Cited by:

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