On the third ABC index of trees and unicyclic graphs

TL;DR

This paper investigates the maximum third atom-bond connectivity index ($ABC_3$) for unicyclic graphs with fixed girth and trees with fixed diameter, providing exact characterizations of the extremal graphs.

Contribution

It determines the maximal $ABC_3$ index for specific classes of graphs and characterizes the extremal structures, advancing understanding of this index in graph theory.

Findings

Identified the maximum $ABC_3$ index for unicyclic graphs with given girth.

Determined the maximum $ABC_3$ index for trees with given diameter.

Characterized the extremal graphs achieving these maxima.

Abstract

Let $G=(V,E)$ be a simple connected graph with vertex set $V(G)$ and edge set $E(G)$. The third atom-bond connectivity index, $ABC_3$ index, of $G$ is defined as $ABC_3(G)=\sum\limits_{uv\in E(G)}\sqrt{\frac{e(u)+e(v)-2}{e(u)e(v)}}$, where eccentricity $e(u)$ is the largest distance between $u$ and any other vertex of $G$, namely $e(u)=\max\{d(u,v)|v\in V(G)\}$. This work determines the maximal $ABC_3$ index of unicyclic graphs with any given girth and trees with any given diameter, and characterizes the corresponding graphs.