Extremal Values of the Atom-Bond Connectivity Index for Trees with Given Roman Domination Numbers

TL;DR

This paper investigates the extremal values of the atom-bond connectivity index in trees with a fixed Roman domination number, establishing bounds and characterizing the extremal tree structures.

Contribution

It introduces bounds for the ABC index in trees based on order and Roman domination number, and characterizes trees achieving these bounds.

Findings

Derived lower and upper bounds for the ABC index in trees.

Identified tree structures that attain extremal ABC index values.

Analyzed the influence of Roman domination number on the ABC index.

Abstract

Consider that $\mathbb{G}=(\mathbb{X}, \mathbb{Y})$ is a simple, connected graph with $\mathbb{X}$ as the vertex set and $\mathbb{Y}$ as the edge set. The atom-bond connectivity ($ABC$) index is a novel topological index that Estrada introduced in Estrada et al. (1998). It is defined as $$ A B C(\mathbb{G})=\sum_{xy \in Y(\mathbb{G})} \sqrt{\frac{\zeta_x+\zeta_y-2}{\zeta_x \zeta_y}} $$ where $\zeta_x$ and $\zeta_x$ represent the degrees of the vertices $x$ and $y$, respectively. In this work, we explore the behavior of the $A B C$ index for tree graphs. We establish both lower and upper bounds for the $A B C$ index, expressed in terms of the graph's order and its Roman domination number. Additionally, we characterize the tree structures that correspond to these extremal values, offering a deeper understanding of how the Roman domination number ($RDN$) influences the $A B C$ index in tree graphs.