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On structural properties of trees with minimal atom-bond connectivity index III: Trees with pendent paths of length three

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  • Dimitrov, Darko
  • Du, Zhibin
  • da Fonseca, Carlos M.

Abstract

The atom-bond connectivity (ABC) index is a degree-based graph topological index that found chemical applications, including those of predicting the stability of alkanes and the strain energy of cycloalkanes. Several structural properties of the trees with minimal ABC index were proved recently, however the complete characterization of the minimal-ABC trees is still an open problem. It is known that minimal-ABC trees can have at most one pendent path of length 3. It is also known that the minimal-ABC trees that have a pendent path of length 3 do not contain so-called Bk-branches, with k ≥ 4, and do not contain more than two B2-branches. Here, we improve the latter result by showing that minimal-ABC trees of order larger than 168 and with a pendent path of length 3 do not contain B2-branches. Moreover, we show that trees with minimal ABC index with a pendent path of length 3 do not contain B1-branches.

Suggested Citation

  • Dimitrov, Darko & Du, Zhibin & da Fonseca, Carlos M., 2016. "On structural properties of trees with minimal atom-bond connectivity index III: Trees with pendent paths of length three," Applied Mathematics and Computation, Elsevier, vol. 282(C), pages 276-290.
    • Handle: RePEc:eee:apmaco:v:282:y:2016:i:c:p:276-290
    DOI: 10.1016/j.amc.2016.02.019
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    Cited by:

    1. Shao, Zehui & Wu, Pu & Gao, Yingying & Gutman, Ivan & Zhang, Xiujun, 2017. "On the maximum ABC index of graphs without pendent vertices," Applied Mathematics and Computation, Elsevier, vol. 315(C), pages 298-312.
    2. Dimitrov, Darko, 2017. "On structural properties of trees with minimal atom-bond connectivity index IV: Solving a conjecture about the pendent paths of length three," Applied Mathematics and Computation, Elsevier, vol. 313(C), pages 418-430.
    3. Dimitrov, Darko & Du, Zhibin, 2021. "A solution of the conjecture about big vertices of minimal-ABC trees," Applied Mathematics and Computation, Elsevier, vol. 397(C).
    4. Lin, Wenshui & Chen, Jianfeng & Wu, Zhixi & Dimitrov, Darko & Huang, Linshan, 2018. "Computer search for large trees with minimal ABC index," Applied Mathematics and Computation, Elsevier, vol. 338(C), pages 221-230.

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