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The weighted vertex PI index of (n,m)-graphs with given diameter

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  • Ma, Gang
  • Bian, Qiuju
  • Wang, Jianfeng

Abstract

The weighted vertex PI index of a graph G is defined byPIw(G)=∑e=uv∈E(G)(dG(u)+dG(v))(nu(e|G)+nv(e|G))where nu(e|G) denotes the number of vertices in G whose distance to the vertex u is smaller than the distance to the vertex v. In this paper, we give the upper bound and the corresponding extremal graphs on the weighted vertex PI index of (n, m)-graphs with diameter d. The lower bound and the corresponding extremal graphs on the first Zagreb index and the weighted vertex PI index of trees with diameter d are given by two procedures. The extremal graphs, given by the two procedures, are also the extremal graphs which attain the lower bound on the first Zagreb index among all connected graphs with n vertices and diameter d.

Suggested Citation

  • Ma, Gang & Bian, Qiuju & Wang, Jianfeng, 2019. "The weighted vertex PI index of (n,m)-graphs with given diameter," Applied Mathematics and Computation, Elsevier, vol. 354(C), pages 329-337.
    • Handle: RePEc:eee:apmaco:v:354:y:2019:i:c:p:329-337
    DOI: 10.1016/j.amc.2019.02.044
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    References listed on IDEAS

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    1. Shi, Yongtang, 2015. "Note on two generalizations of the Randić index," Applied Mathematics and Computation, Elsevier, vol. 265(C), pages 1019-1025.
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    Cited by:

    1. Wanping Zhang & Jixiang Meng & Baoyindureng Wu, 2022. "The upper bounds on the Steiner k-Wiener index in terms of minimum and maximum degrees," Journal of Combinatorial Optimization, Springer, vol. 44(2), pages 1199-1220, September.

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