investigation of closed derivation formulas for gq and qg indices of a graph via m-polynomial

A topological index is a numerical data which significantly correlates with the fundamental topology of a given chemical structure. the m-polynomial is a key mathematical tool to determine the degree-dependent topological indices. very recently, the geometric-quadratic (gq) and quadratic-geometric (qg) indices of a graph are introduced and computed their values by their respective mathematical formulas on some standard graphs and jagged-rectangle benzenoid system. in this research work, we propose m-polynomial based closed derivation formulas for determining the above two indices. in addition, we derive the gq and qg indices for each of the abovementioned graphs by applying the derivation formulas, and also produce some fundamental relationships between the indices.