On the Difference of Atom-Bond Sum-Connectivity and Atom-Bond-Connectivity Indices

TL;DR

This paper investigates the difference between the atom-bond sum-connectivity (ABS) and atom-bond connectivity (ABC) indices, showing positivity under certain conditions and computing differences for small trees.

Contribution

It establishes the positivity of the ABS-ABC difference for specific classes of graphs and line graphs, and provides computational data for small trees.

Findings

ABS-ABC is positive for graphs with minimum degree ≥ 2.

The difference is positive for line graphs of certain graphs, excluding paths and cycles.

Computed differences for all trees up to order 15.

Abstract

The atom-bond-connectivity (ABC) index is one of the well-investigated degree-based topological indices. The atom-bond sum-connectivity (ABS) index is a modified version of the ABC index, which was introduced recently. The primary goal of the present paper is to investigate the difference between the aforementioned two indices, namely $ABS-ABC$. It is shown that the difference $ABS-ABC$ is positive for all graphs of minimum degree at least $2$ as well as for all line graphs of those graphs of order at least $5$ that are different from the path and cycle graphs. By means of computer search, the difference $ABS-ABC$ is also calculated for all trees of order at most $15$.