Extremal Bounds of the Atom-Bond Connectivity Index in Trees with a Fixed Roman Domination Number

Let G = (X, Y) be a simple, connected graph, where X denotes the set of vertices and Y represents the set of edges. The atom-bond connectivity (ABC) index, introduced by Estrada et al. [10], is a topological descriptor used in mathematical chemistry. It is given by, ABC(G) =Xxy∈Ysζx + ζy − 2ζxζy, where ζx and ζy 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.