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Extremal Values of the Atom-Bond Connectivity Index for Trees with Given Roman Domination Numbers https://arxiv.org/abs/2411.11850 #mathGM

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

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 $$ A B C(\mathbb{G})=\sum_{xy \in Y(\mathbb{G})} \sqrt{\frac{ζ_x+ζ_y-2}{ζ_x ζ_y}} $$ where $ζ_x$ and $ζ_x$ 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.

arXiv.org
November 21, 2024 at 3:10 AM · · feed2toot · 0 · 0 · 0
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