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Computer search for large trees with minimal ABC index

Author

Listed:
  • Lin, Wenshui
  • Chen, Jianfeng
  • Wu, Zhixi
  • Dimitrov, Darko
  • Huang, Linshan

Abstract

The atom-bond connectivity (ABC) index of a graph G = (V, E) is defined as ABC(G)=∑vivj∈E(di+dj−2)/(didj), where V = {v0,v1,⋅⋅⋅, vn − 1} and di denotes the degree of vertex vi of G. This molecular structure descriptor found interesting applications in chemistry, and has become one of the most actively studied vertex-degree-based graph invariants. However, the problem of characterizing n-vertex tree(s) with minimal ABC index remains open and was coined as the “ABC index conundrum”. In attempts to guess the general structure of such trees, several computer search algorithms were developed and tested up to n = 800. However, for large n, all current search programs seem too powerless. For example, the fastest one up to date reported recently in [30] costs 2.2 h for n = 800 on a single PC with two CPU cores. In this paper, we significantly refine the known features of the degree sequence of a tree with minimal ABC index. With the refined features a search program was implemented with OpenMP. Our program was tested on a single PC with 4 CPU cores, and identified all n-vertex tree(s) with minimal ABC index up to n = 1100 within 207.1 h. Some observations are made based on the search results, which indicate some possible directions in further investigation of the problem of characterizing n-vertex tree(s) with minimal ABC index.

Suggested Citation

  • 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.
  • Handle: RePEc:eee:apmaco:v:338:y:2018:i:c:p:221-230
    DOI: 10.1016/j.amc.2018.06.012
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    References listed on IDEAS

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    1. Jinsong Chen & Jianping Liu & Qiaoliang Li, 2013. "The Atom-Bond Connectivity Index of Catacondensed Polyomino Graphs," Discrete Dynamics in Nature and Society, Hindawi, vol. 2013, pages 1-7, March.
    2. Palacios, José Luis, 2017. "Bounds for the augmented Zagreb and the atom-bond connectivity indices," Applied Mathematics and Computation, Elsevier, vol. 307(C), pages 141-145.
    3. 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.
    4. 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.
    5. 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.
    6. Gao, Wei & Farahani, Mohammad Reza & Wang, Shaohui & Husin, Mohamad Nazri, 2017. "On the edge-version atom-bond connectivity and geometric arithmetic indices of certain graph operations," Applied Mathematics and Computation, Elsevier, vol. 308(C), pages 11-17.
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    Cited by:

    1. Chen, Xiaodan & Li, Xiuyu & Lin, Wenshui, 2021. "On connected graphs and trees with maximal inverse sum indeg index," Applied Mathematics and Computation, Elsevier, vol. 392(C).
    2. 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).
    3. Yu Yang & Long Li & Wenhu Wang & Hua Wang, 2020. "On BC-Subtrees in Multi-Fan and Multi-Wheel Graphs," Mathematics, MDPI, vol. 9(1), pages 1-29, December.

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