Remarks on the maximum atom-bond connectivity index of graphs with given parameters

TL;DR

This paper investigates the maximum atom-bond connectivity (ABC) index in graphs with specific parameters, confirming conjectures about their structure for certain cases, which aids in understanding molecular descriptors.

Contribution

It proves the structure of graphs with edge-connectivity one that maximize the ABC index and supports a conjecture about graphs with fixed chromatic number, advancing theoretical understanding.

Findings

Graphs with edge-connectivity one that maximize ABC are characterized.

Confirmed the conjecture for graphs with fixed chromatic number when order is divisible by that number.

Provides structural insights into graphs optimizing the ABC index.

Abstract

The atom-bond connectivity (ABC) index is a degree-based molecular structure descriptor that can be used for modelling thermodynamic properties of organic chemical compounds. Motivated by its applicable potential, a series of investigations have been carried out in the past several years. In this note we first consider graphs with given edge-connectivity that attain the maximum ABC index. In particular, we give an affirmative answer to the conjecture about the structure of graphs with edge-connectivity equal to one that maximize the ABC index, which was recently raised by Zhang, Yang, Wang and Zhang~\cite{zywz mabciggp-2016}. In addition, we provide supporting evidence for another conjecture posed by the same authors which concerns graphs that maximize the ABC index among all graphs with chromatic number equal to some fixed $\chi \geq 3$. Specifically, we confirm this conjecture in the case where the order of the graph is divisible by $\chi$.