On the third ABC index of trees and unicyclic graphs

TL;DR

This work determines extremal values of the third atom-bond connectivity index $ABC_3$ for two graph families: unicyclic graphs with fixed girth and trees with fixed diameter. By structurally characterizing extremal graphs as leaf-augmented cycles $H(n,k;n_1,\dots,n_k)$ and employing the edge-contribution function $f(x,y)$, the authors derive sharp upper bounds and complete characterizations of extremal cases. The global maximum over unicyclic graphs of order $n$ is shown to be $n\sqrt{\frac{1}{2}}$, attained by specific $H(n,k;\cdots)$ configurations; the second maximum is precisely described via $H^t$ and related constructions with explicit conditions. For trees, the maximal $ABC_3$ is achieved by caterpillars concentrated at the diameter center, with explicit formulas depending on the diameter and a secondary maximizer given by $C_{n,4}$. Overall, the paper provides exact extremal structures and closed-form bounds for $ABC_3$, enhancing its applicability in chemical graph theory and network analysis.

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.

Theorems & Definitions (20)

• Lemma 2.1: 14
• Lemma 2.2: 15
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Theorem 2.1
• proof