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On maximum Wiener index of trees and graphs with given radius

Author

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  • Kinkar Ch. Das

    (Sungkyunkwan University)

  • M. J. Nadjafi-Arani

    (Mahallat Institute of Higher Education)

Abstract

Let G be a connected graph of order n. The long-standing open and close problems in distance graph theory are: what is the Wiener index W(G) or average distance $$\mu (G)$$ μ ( G ) among all graphs of order n with diameter d (radius r)? There are very few number of articles where were worked on the relationship between radius or diameter and Wiener index. In this paper, we give an upper bound on Wiener index of trees and graphs in terms of number of vertices n, radius r, and characterize the extremal graphs. Moreover, from this result we give an upper bound on $$\mu (G)$$ μ ( G ) in terms of order and independence number of graph G. Also we present another upper bound on Wiener index of graphs in terms of number of vertices n, radius r and maximum degree $$\Delta $$ Δ , and characterize the extremal graphs.

Suggested Citation

  • Kinkar Ch. Das & M. J. Nadjafi-Arani, 2017. "On maximum Wiener index of trees and graphs with given radius," Journal of Combinatorial Optimization, Springer, vol. 34(2), pages 574-587, August.
    • Handle: RePEc:spr:jcomop:v:34:y:2017:i:2:d:10.1007_s10878-016-0092-y
    DOI: 10.1007/s10878-016-0092-y
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    References listed on IDEAS

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    1. Das, Kinkar Ch. & Gutman, Ivan & Nadjafi–Arani, Mohammad J., 2015. "Relations between distance–based and degree–based topological indices," Applied Mathematics and Computation, Elsevier, vol. 270(C), pages 142-147.
    2. Guifu Su & Liming Xiong & Xiaofeng Su & Xianglian Chen, 2015. "Some results on the reciprocal sum-degree distance of graphs," Journal of Combinatorial Optimization, Springer, vol. 30(3), pages 435-446, October.
    3. Baoyindureng Wu & Xinhui An & Guojie Liu & Guiying Yan & Xiaoping Liu, 2013. "Minimum degree, edge-connectivity and radius," Journal of Combinatorial Optimization, Springer, vol. 26(3), pages 585-591, October.
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    Citations

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    Cited by:

    1. Hongfang Liu & Jinxia Liang & Yuhu Liu & Kinkar Chandra Das, 2023. "A Combinatorial Approach to Study the Nordhaus–Guddum-Type Results for Steiner Degree Distance," Mathematics, MDPI, vol. 11(3), pages 1-19, February.
    2. Xiangxiang Liu & Ligong Wang & Xihe Li, 2020. "The Wiener index of hypergraphs," Journal of Combinatorial Optimization, Springer, vol. 39(2), pages 351-364, February.
    3. Kexiang Xu & Haiqiong Liu & Kinkar Ch. Das & Sandi Klavžar, 2018. "Embeddings into almost self-centered graphs of given radius," Journal of Combinatorial Optimization, Springer, vol. 36(4), pages 1388-1410, November.
    4. Debarun Ghosh & Ervin Győri & Addisu Paulos & Nika Salia & Oscar Zamora, 2020. "The maximum Wiener index of maximal planar graphs," Journal of Combinatorial Optimization, Springer, vol. 40(4), pages 1121-1135, November.
    5. Hamid Darabi & Yaser Alizadeh & Sandi Klavžar & Kinkar Chandra Das, 2021. "On the relation between Wiener index and eccentricity of a graph," Journal of Combinatorial Optimization, Springer, vol. 41(4), pages 817-829, May.

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