In [1], Hamzeh, Iranmanesh and Hossein-Zadeh and M. V. Diudea recently introduced the generalized degree distance of graphs. In this paper, we present explicit formulas for this new graph invariant of the Cartesian product, composition, join, disjunction and symmetric difference of graphs and introduce generalized and modified generalized degree distance polynomials of graphs, such that their first derivatives at x = 1 are respectively, equal to the generalized degree distance and the modified generalized degree distance. These polynomials are related to Wiener-type invariant polynomial of graphs.