arXiv 1706.028912017-06math.CO

给定叶数的ABC-极小树的结构

The structure of ABC-minimal trees with given number of leaves

Bojan Mohar

备注：: 16 pages
发表信息：: MATCH Commun. Math. Comput. Chem. 79 (2018) 415-430

摘要：

原子-键连接性（ABC）指数是一种基于度的分子描述符，具有多种化学应用。Lin et al. [W. Lin，J. Chen，C.马，Y。Zhang，J. Chen，D. Zhang和F.贾，关于给定叶数树中具有最小ABC指标的树，MATCH通讯。数学。计算。76（2016）131-140]提出了关于具有固定数量的$t$叶的树的最小可能ABC指数的猜想。我们表明，这个猜想是不正确的，我们证明了正确的答案是什么。结果表明，$T_t$是$t\ge 1195$的唯一极值树，它的阶数为$|T_t| = t + \lfloor \tfrac{t}{10}\rfloor +1$（当$t$ mod 10在0和4之间或当$t$ mod 10为5、6或7且$t$足够大时）或$|T_t| = t + \lfloor \tfrac{t}{10}\rfloor + 2$（当$t$ mod 10为8或9或当$t$ mod 10为5、6或7且$t$足够小时），其ABC索引为$( \sqrt{\tfrac{10}{11}} + \tfrac{1}{10}\sqrt{\tfrac{1}{11}} )t + O(1)$ .

英文摘要：

The atom-bond connectivity (ABC) index is a degree-based molecular descriptor with diverse chemical applications. Recent work of Lin et al. [W. Lin, J. Chen, C. Ma, Y. Zhang, J. Chen, D. Zhang, and F. Jia, On trees with minimal ABC index among trees with given number of leaves, MATCH Commun. Math. Comput. Chem. 76 (2016) 131-140] gave rise to a conjecture about the minimum possible ABC-index of trees with a fixed number $t$ of leaves. We show that this conjecture is incorrect and we prove what the correct answer is. It is shown that the extremal tree $T_t$ is unique for $t\ge 1195$, it has order $|T_t| = t + \lfloor \tfrac{t}{10}\rfloor +1$ (when $t$ mod 10 is between 0 and 4 or when it is 5, 6, or 7 and $t$ is sufficiently large) or $|T_t| = t + \lfloor \tfrac{t}{10}\rfloor + 2$ (when $t$ mod 10 is 8 or 9 or when it is 5, 6, or 7 and $t$ is sufficiently small) and its ABC-index is $( \sqrt{\tfrac{10}{11}} + \tfrac{1}{10}\sqrt{\tfrac{1}{11}} )t + O(1)$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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