General Neighborhood Multiplicative Zagreb Indices: Extremal Results and Structural Characterization of Molecular Trees

Degree-based topological indices play a central role in characterizing graph structures and their chemical applications. Among these, multiplicative Zagreb indices have attracted considerable attention due to their strong discriminative power and relevance in chemical graph theory. Neighborhood versions of these indices extend the classical concept by incorporating the aggregate degree information of adjacent vertices, capturing more subtle structural effects related to local branching. Trees, as connected acyclic graphs, provide a natural and tractable class for studying the extremal behaviors of these indices, while molecular trees—trees with a maximum degree of at most four—serve as chemically meaningful models of acyclic organic compounds. Investigating extremal values on these structures offers both theoretical insight into the indices’ behavior and identification of molecular graphs that maximize or minimize them. In this work, we determine the maximal and minimal values of the neighborhood-based multiplicative Zagreb indices for trees of fixed order and prescribed maximum degree, and we provide a complete structural characterization of all extremal graphs. Special attention is given to molecular trees, for which explicit extremal bounds are derived and all optimal structures are identified. These results provide efficient tools for evaluating the indices and illuminate the structural principles governing their extremal behavior.