Efficient computation of trees with minimal atom-bond connectivity index

TL;DR

This paper introduces an efficient method for computing trees with the minimal atom-bond connectivity (ABC) index, a key molecular descriptor, challenging existing conjectures and proposing new ones based on degree sequence analysis.

Contribution

It presents a novel approach to efficiently identify trees with minimal ABC index using degree sequences and graph properties, advancing understanding in molecular graph theory.

Findings

Disproved some existing conjectures about minimal ABC trees

Proposed new conjectures based on degree sequence analysis

Provided an efficient computational method for minimal ABC trees

Abstract

The {\em atom-bond connectivity (ABC) index} is one of the recently most investigated degree-based molecular structure descriptors, that have applications in chemistry. For a graph $G$, the ABC index is defined as $\sum_{uv\in E(G)}\sqrt{\frac{(d(u) +d(v)-2)}{d(u)d(v)}}$, where $d(u)$ is the degree of vertex $u$ in $G$ and $E(G)$ is the set of edges of $G$. Despite many attempts in the last few years, it is still an open problem to characterize trees with minimal $ABC$ index. In this paper, we present an efficient approach of computing trees with minimal ABC index, by considering the degree sequences of trees and some known properties of the graphs with minimal $ABC$ index. The obtained results disprove some existing conjectures end suggest new ones to be set.