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Abstract

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.