Complete characterization of the minimal-ABC trees

TL;DR

This paper solves the open problem of characterizing minimal-ABC trees by proving a conjecture and providing a complete description of their structure, enabling exact determination for any size.

Contribution

It confirms a conjecture about the structure of large minimal-ABC trees and offers a method to precisely identify minimal-ABC trees of any order.

Findings

Proof that large minimal-ABC trees consist of a root and specific branches

Reduction of the search space for minimal-ABC trees

Exact determination of minimal-ABC trees for any order

Abstract

The problem of characterizing trees with minimal atom-bond-connectivity index (minimal-ABC trees) has a reputation as one of the most demanding recent open optimization problems in mathematical chemistry. Here firstly, we give an affirmative answer to the conjecture, which states that enough large minimal-ABC trees are comprised solely of a root vertex and so-called $D_z$- and $D_{z+1}$-branches. Based on the presented theoretical results here and some already known results, we obtain enough constraints to reduce the search space and solve the optimization problem, and thus, determine exactly the minimal-ABC trees of a given arbitrary order.