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Minimizing Kirchhoff index among graphs with a given vertex bipartiteness

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  • Liu, Jia-Bao
  • Pan, Xiang-Feng

Abstract

The resistance distance between any two vertices of a graph G is defined as the effective resistance between them if each edge of G is replaced by a unit resistor. The Kirchhoff index Kf(G) is the sum of the resistance distances between all the pairs of vertices in G. The vertex bipartiteness vb of a graph G is the minimum number of vertices whose deletion from G results in a bipartite graph. In this paper, we characterize the graph having the minimum Kf(G) values among graphs with a fixed number n of vertices and fixed vertex bipartiteness, 1≤vb≤n−3.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:291:y:2016:i:c:p:84-88
    DOI: 10.1016/j.amc.2016.06.017
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    References listed on IDEAS

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    1. Liu, Jia-Bao & Pan, Xiang-Feng, 2015. "Asymptotic incidence energy of lattices," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 422(C), pages 193-202.
    2. Liu, Jia-Bao & Pan, Xiang-Feng, 2015. "A unified approach to the asymptotic topological indices of various lattices," Applied Mathematics and Computation, Elsevier, vol. 270(C), pages 62-73.
    3. Liu, Jia-Bao & Pan, Xiang-Feng & Hu, Fu-Tao & Hu, Feng-Feng, 2015. "Asymptotic Laplacian-energy-like invariant of lattices," Applied Mathematics and Computation, Elsevier, vol. 253(C), pages 205-214.
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    Cited by:

    1. Sajjad, Wasim & Sardar, Muhammad Shoaib & Pan, Xiang-Feng, 2024. "Computation of resistance distance and Kirchhoff index of chain of triangular bipyramid hexahedron," Applied Mathematics and Computation, Elsevier, vol. 461(C).
    2. Liu, Jia-Bao & Zhao, Jing & Cai, Zheng-Qun, 2020. "On the generalized adjacency, Laplacian and signless Laplacian spectra of the weighted edge corona networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 540(C).
    3. Praba, B. & Saranya, R., 2020. "Application of the graph cellular automaton in generating languages," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 168(C), pages 111-121.
    4. Wang, Daohua & Zeng, Cheng & Zhao, Zixuan & Wu, Zhiqiang & Xue, Yumei, 2023. "Kirchhoff index of a class of polygon networks," Chaos, Solitons & Fractals, Elsevier, vol. 168(C).
    5. Faxu Li & Hui Xu & Liang Wei & Defang Wang, 2023. "RETRACTED ARTICLE: Identifying vital nodes in hypernetwork based on local centrality," Journal of Combinatorial Optimization, Springer, vol. 45(1), pages 1-13, January.
    6. Jian Lu & Shu-Bo Chen & Jia-Bao Liu & Xiang-Feng Pan & Ying-Jie Ji, 2019. "Further Results on the Resistance-Harary Index of Unicyclic Graphs," Mathematics, MDPI, vol. 7(2), pages 1-13, February.
    7. Sardar, Muhammad Shoaib & Pan, Xiang-Feng & Xu, Si-Ao, 2020. "Computation of resistance distance and Kirchhoff index of the two classes of silicate networks," Applied Mathematics and Computation, Elsevier, vol. 381(C).
    8. Fang Gao & Xiaoxin Li & Kai Zhou & Jia-Bao Liu, 2018. "The Extremal Graphs of Some Topological Indices with Given Vertex k -Partiteness," Mathematics, MDPI, vol. 6(11), pages 1-11, November.
    9. Jia Wei & Jing Wang, 2022. "Spectra of Complemented Triangulation Graphs," Mathematics, MDPI, vol. 10(17), pages 1-9, September.
    10. 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.
    11. Wenyu Shi & Qiang Tang, 2023. "RETRACTED ARTICLE: Cost-optimized data placement strategy for social network with security awareness in edge-cloud computing environment," Journal of Combinatorial Optimization, Springer, vol. 45(1), pages 1-15, January.
    12. Jia-Bao Liu & S. N. Daoud, 2019. "Number of Spanning Trees in the Sequence of Some Graphs," Complexity, Hindawi, vol. 2019, pages 1-22, March.
    13. 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.
    14. Jia-Bao Liu & Muhammad Kashif Shafiq & Haidar Ali & Asim Naseem & Nayab Maryam & Syed Sheraz Asghar, 2019. "Topological Indices of m th Chain Silicate Graphs," Mathematics, MDPI, vol. 7(1), pages 1-16, January.
    15. Li Zhang & Jing Zhao & Jia-Bao Liu & Salama Nagy Daoud, 2019. "Resistance Distance in the Double Corona Based on R -Graph," Mathematics, MDPI, vol. 7(1), pages 1-13, January.
    16. Li Zhang & Jing Zhao & Jia-Bao Liu & Micheal Arockiaraj, 2018. "Resistance Distance in H -Join of Graphs G 1 , G 2 , … , G k," Mathematics, MDPI, vol. 6(12), pages 1-10, November.
    17. Hong, Yunchao & Zhu, Zhongxun & Luo, Amu, 2018. "Some transformations on multiplicative eccentricity resistance-distance and their applications," Applied Mathematics and Computation, Elsevier, vol. 323(C), pages 75-85.
    18. 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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