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

TL;DR

This paper corrects a previous conjecture about the minimal ABC-index trees with a fixed number of leaves, providing the exact structure, size, and ABC-index for these extremal trees.

Contribution

It disproves a prior conjecture and establishes the precise structure and properties of ABC-minimal trees with given leaves, including their size and uniqueness for large t.

Findings

The conjecture was incorrect.

The extremal tree T_t is unique for t ≥ 1195.

The size of T_t is approximately t + t/10 + 1 or 2, depending on t mod 10.

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 $|T_t| = t + \lfloor \tfrac{t}{10}\rfloor +1$ (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 $( \sqrt{\tfrac{10}{11}} + \tfrac{1}{10}\sqrt{\tfrac{1}{11}} )t + O(1)$.