# Forbidden branches in trees with minimal atom-bond connectivity index

## TL;DR

This paper investigates the structural properties of trees with the minimal atom-bond connectivity (ABC) index, revealing restrictions on certain branch configurations to advance their complete characterization in chemical graph theory.

## Contribution

It establishes new structural constraints on minimal-ABC trees, specifically ruling out the coexistence of certain branch types, thus progressing toward their full classification.

## Key findings

- Minimal-ABC trees cannot contain both a B4-branch and B1 or B2-branches simultaneously.
- Provides new structural properties that limit the configurations of minimal-ABC trees.
- Advances understanding of the structure of trees with minimal atom-bond connectivity index.

## Abstract

The atom-bond connectivity (ABC) index has been, in recent years, one of the most actively studied vertex-degree-based graph invariants in chemical graph theory. For a given graph $G$, the ABC index is defined as $\sum_{uv\in E}\sqrt{\frac{d(u) +d(v)-2}{d(u)d(v)}}$, where $d(u)$ is the degree of vertex $u$ in $G$ and $E(G)$ denotes the set of edges of $G$. In this paper we present some new structural properties of trees with a minimal ABC index (also refer to as a minimal-ABC tree), which is a step further towards understanding their complete characterization. We show that a minimal-ABC tree cannot simultaneously contain a $B_4$-branch and $B_1$ or $B_2$-branches.