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Solution to a conjecture on a Nordhaus–Gaddum type result for the Kirchhoff index

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  • Yang, Yujun
  • Cao, Yuliang
  • Yao, Haiyuan
  • Li, Jing

Abstract

Let G be a connected graph. The resistance distance between any two vertices of G is defined as the net effective resistance between them if each edge of G is replaced by a unit resistor. The Kirchhoff index of G, denoted by Kf(G), is the sum of resistance distances between all pairs of vertices in G. In [28], it was conjectured that for a connected n-vertex graph G with a connected complement G¯,Kf(G)+Kf(G¯)≤n3−n6+n∑k=1n−11n−4sin2kπ2n,with equality if and only if G or G¯ is the path graph Pn. In this paper, by employing combinatorial and electrical techniques, we show that the conjecture is true except for a complementary pair of small graphs on five vertices.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:332:y:2018:i:c:p:241-249
    DOI: 10.1016/j.amc.2018.03.070
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    References listed on IDEAS

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    1. Jiang, Zhuozhuo & Yan, Weigen, 2017. "Resistance between two nodes of a ring network," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 484(C), pages 21-26.
    2. 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.
    3. 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.
    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.
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    Cited by:

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    4. 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.

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