# On structural properties of trees with minimal atom-bond connectivity index IV: Solving a conjecture about the pendent paths of length three

## TL;DR

This paper proves that for large trees (order > 415), the minimal atom-bond connectivity index trees cannot have pendent paths of length three, advancing understanding of their structural properties.

## Contribution

It confirms a conjecture that trees with minimal ABC index lack pendent paths of length three when their order exceeds 415.

## Key findings

- Trees with minimal ABC index have no pendent path of length three if order > 415.
- Supports the conjecture that such paths are absent in large minimal-ABC trees.
- Provides a strengthened version of the previous conjecture with a tighter bound.

## Abstract

The atom-bond connectivity (ABC) index is one of the most investigated degree-based molecular structure descriptors with a variety of chemical applications. It is known that among all connected graphs, the trees minimize the ABC index. However, a full characterization of trees with a minimal ABC index is still an open problem. By now, one of the proved properties is that a tree with a minimal ABC index may have, at most, one pendent path of length $3$, with the conjecture that it cannot be a case if the order of a tree is larger than $1178$. Here, we provide an affirmative answer of a strengthened version of that conjecture, showing that a tree with minimal ABC index cannot contain a pendent path of length $3$ if its order is larger than $415$.