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The Nordhaus–Gaddum-type inequality for the Wiener polarity index

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  • Zhang, Yanhong
  • Hu, Yumei

Abstract

The Wiener polarity index Wp(G) of a graph G is the number of unordered pairs of vertices {u, v} of G such that the distance between u and v is 3. In this paper, we study the Nordhaus–Gaddum-type inequality for the Wiener polarity index of a graph G of order n. Due to concerns that both Wp(G) and Wp(G¯) are nontrivial only when diam(G)=3 and diam(G¯)=3, we firstly consider the crucial case and get that 2≤Wp(G)+Wp(G¯)≤⌈n2⌉⌊n2⌋−n+2. Moreover, the bounds are best possible, and the corresponding extremal graphs are also presented. Then we generalize the results to all connected simple graphs. Furthermore, we discuss the Nordhaus–Gaddum-type inequality for trees and unicyclic graphs, respectively.

Suggested Citation

  • Zhang, Yanhong & Hu, Yumei, 2016. "The Nordhaus–Gaddum-type inequality for the Wiener polarity index," Applied Mathematics and Computation, Elsevier, vol. 273(C), pages 880-884.
    • Handle: RePEc:eee:apmaco:v:273:y:2016:i:c:p:880-884
    DOI: 10.1016/j.amc.2015.10.045
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    Citations

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    Cited by:

    1. Lei, Hui & Li, Tao & Ma, Yuede & Wang, Hua, 2018. "Analyzing lattice networks through substructures," Applied Mathematics and Computation, Elsevier, vol. 329(C), pages 297-314.
    2. Gutman, Ivan, 2016. "On Steiner degree distance of trees," Applied Mathematics and Computation, Elsevier, vol. 283(C), pages 163-167.
    3. Hua, Hongbo & Das, Kinkar Ch., 2016. "On the Wiener polarity index of graphs," Applied Mathematics and Computation, Elsevier, vol. 280(C), pages 162-167.
    4. Ashrafi, Ali Reza & Ghalavand, Ali, 2017. "Ordering chemical trees by Wiener polarity index," Applied Mathematics and Computation, Elsevier, vol. 313(C), pages 301-312.
    5. Noureen, Sadia & Bhatti, Akhlaq Ahmad & Ali, Akbar, 2021. "Towards the solution of an extremal problem concerning the Wiener polarity index of alkanes," Chaos, Solitons & Fractals, Elsevier, vol. 144(C).
    6. Ali, Akbar & Du, Zhibin & Ali, Muhammad, 2018. "A note on chemical trees with minimum Wiener polarity index," Applied Mathematics and Computation, Elsevier, vol. 335(C), pages 231-236.

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