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On the ordering of distance-based invariants of graphs

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  • Liu, Muhuo
  • Das, Kinkar Ch.

Abstract

Let d(u, v) be the distance between u and v of graph G, and let Wf(G) be the sum of f(d(u, v)) over all unordered pairs {u, v} of vertices of G, where f(x) is a function of x. In some literatures, Wf(G) is also called the Q-index of G. In this paper, some unified properties to Q-indices are given, and the majorization theorem is illustrated to be a good tool to deal with the ordering problem of Q-index among trees with n vertices. With the application of our new results, we determine the four largest and three smallest (resp. four smallest and three largest) Q-indices of trees with n vertices for strictly decreasing (resp. increasing) nonnegative function f(x), and we also identify the twelve largest (resp. eighteen smallest) Harary indices of trees of order n ≥ 22 (resp. n ≥ 38) and the ten smallest hyper-Wiener indices of trees of order n ≥ 18, which improve the corresponding main results of Xu (2012) and Liu and Liu (2010), respectively. Furthermore, we obtain some new relations involving Wiener index, hyper-Wiener index and Harary index, which gives partial answers to some problems raised in Xu (2012).

Suggested Citation

  • Liu, Muhuo & Das, Kinkar Ch., 2018. "On the ordering of distance-based invariants of graphs," Applied Mathematics and Computation, Elsevier, vol. 324(C), pages 191-201.
    • Handle: RePEc:eee:apmaco:v:324:y:2018:i:c:p:191-201
    DOI: 10.1016/j.amc.2017.12.015
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    References listed on IDEAS

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    1. Dehmer, Matthias & Emmert-Streib, Frank & Shi, Yongtang, 2015. "Graph distance measures based on topological indices revisited," Applied Mathematics and Computation, Elsevier, vol. 266(C), pages 623-633.
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