The evolution of the structure of ABC-minimal trees

TL;DR

This paper characterizes the exact structure of trees that minimize the atom-bond connectivity index for large numbers of vertices, revealing their structural evolution and specific degree constraints.

Contribution

It provides the first precise description of ABC-minimal trees for large sizes and analyzes how their structure evolves with increasing vertices.

Findings

The radius of extremal trees is at most 5.

Almost all vertices have degree 1, 2, 4, or 53.

The minimal ABC-index grows linearly with the number of vertices, approximately 0.6774 times n.

Abstract

The atom-bond connectivity (ABC) index is a degree-based molecular descriptor that found diverse chemical applications. Characterizing trees with minimum ABC-index remained an elusive open problem even after serious attempts and is considered by some as one of the most intriguing open problems in mathematical chemistry. In this paper, we describe the exact structure of the extremal trees with sufficiently many vertices and we show how their structure evolves when the number of vertices grows. An interesting fact is that their radius is at most~$5$ and that all vertices except for one have degree at most 54. In fact, all but at most $O(1)$ vertices have degree 1, 2, 4, or 53. Let $\gamma_n = \min\{\abc(T) : T \text{ is a tree of order } n\}$. It is shown that $\gamma_n = \tfrac{1}{365} \sqrt{\tfrac{1}{53}} \Bigl(1 + 26\sqrt{55} + 156\sqrt{106} \Bigr) n + O(1) \approx 0.67737178\, n + O(1)$.