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

Abstract

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 $|T_t|=t+\lfloor \frac{t}{10}\rfloor +1$ (when $t$ mod 10 is between 0 and 4 or when it is 5, 6, or 7 and $t$ is sufficiently large) or $|T_t|=t+\lfloor \frac{t}{10}\rfloor +2$ (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 $(\sqrt{\frac{10}{11}}+\frac{1}{10}\sqrt{\frac{1}{11}})t+O(1)$.

16 pages