A Lower Bound for the First Hyper-Zagreb Index of Trees with given Roman Domination Number

In graph theory, the first Hyper-Zagreb index HM₁(G) is calculated by summing the squares of the degrees of adjacent vertices u and v in molecular graphs. A Roman dominating function (RDF) on a graph G is a function z : V(G) → {0, 1, 2}, where V(G) is the vertex set, with the requirement that for each vertex v with z(v) = 0, there exists an adjacent vertex u such that z(u) = 2. The Roman domination number (RDN) denoted as ζR(G) and represents as the minimum total weight of all vertices under an RDF, and it plays a significant role in network analysis. In this paper, we present a new lower bound for the HZ₁(T ) for trees T with order n and ζR(T ). These findings enhance our understanding of tree structures, providing chemists with a valuable tool for analyzing molecular stability and reactivity. By establishing mathematical bounds on the HZ₁(T ), this research supports more precise predictions of molecular properties and aids in efficient experimental planning in chemical graph theory.