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 $$ABC(\mathbb{G})=\sum _{xy\in Y(\mathbb{G})}\sqrt{\frac{{\zeta }_x+{\zeta }_y-2}{{\zeta }_x{\zeta }_y}}$$ 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 $ABC$ index for tree graphs. We establish both lower and upper bounds for the $ABC$ 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 $ABC$ index in tree graphs.