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On the minimum Kirchhoff index of graphs with a given vertex k-partiteness and edge k-partiteness

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  • He, Weihua
  • Li, Hao
  • Xiao, Shuofa

Abstract

The Kirchhoff index of a connected graph is the sum of the resistance distance between all unordered pairs of vertices and may also be expressed by its Laplacian eigenvalues. The vertex (resp. edge) k-partiteness of a graph G with n vertices is the minimum number of vertices (resp. edges) whose deletion from G yields a k-partite graph. In this paper, we determine the minimum Kirchhoff index of graphs with a given vertex k-partiteness and the minimum Kirchhoff index of graphs with a given edge bipartiteness, when the given edge bipartiteness is no more than n4.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:315:y:2017:i:c:p:313-318
    DOI: 10.1016/j.amc.2017.07.067
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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. 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. 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.
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    Cited by:

    1. 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.
    2. 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.
    3. Shuchao Li & Licheng Zhang & Minjie Zhang, 2019. "On the extremal cacti of given parameters with respect to the difference of zagreb indices," Journal of Combinatorial Optimization, Springer, vol. 38(2), pages 421-442, August.
    4. 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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