Malaysian Journal of Mathematical Sciences, December 2025, Vol. 19, No. 4
Extremal Bounds of the Atom-Bond Connectivity Index in Trees with a Fixed Roman Domination Number
Ali, W., Husin, M. N., and Nadeem, M. F.
Corresponding Email: nazri.husin@umt.edu.my
Received date: 20 February 2025
Accepted date: 25 April 2025
Abstract:
Let $\mathbb{G} = (\mathbb{X}, \mathbb{Y})$ be a simple, connected graph, where $\mathbb{X}$ denotes the set of vertices and $\mathbb{Y}$ represents the set of edges. The atom-bond connectivity $(ABC)$ index, introduced by Estrada et al. \cite{estrada1998atom}, is a topological descriptor used in mathematical chemistry. It is given by,
\begin{align*}
ABC(\mathbb{G}) = \sum_{xy \in \mathbb{Y}} \sqrt{\dfrac{\zeta_x + \zeta_y - 2}{\zeta_x \zeta_y}},
\end{align*}
where $\zeta_{x}$ and $\zeta_{y}$ are the degrees of the vertices $x$ and $y$, respectively. This study investigates the behavior of the $ABC$ index in tree graphs, deriving both upper and lower bounds in terms of the graph's order and its Roman domination number $(RDN)$. Moreover, we identify the specific tree structures that attain these extremal values, providing insights into how the $RDN$ influences the $ABC$ index.
Keywords: $ABC$ index; Roman domination number; tree; chemical graph theory; extremal bounds
