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The numerical coding of the molecular structures on the bases of topological indices plays an important role in the subject of Cheminformatics which is a combination of Computer, Chemistry, and Information Science. In 1972, it was shown that the total π-electron energy of a molecular graph depends upon its structure and it can be obtained by the sum of the square of degrees of the vertices of a molecular graph. Later on, this sum was named as the first Zagreb index. In 2005, for γεR - {0, 1}, the first general Zagreb index of a graph G is defined as M^γ(G) = Σ_{vεV(G)}[d_G(v)]^γ, where d_G(v) is degree of the vertex v in G. In this paper, for each γεR - {0, 1}, we study the first general Zagreb index of the cartesian product of two graphs such that one of the graphs is D-sum graph and the other is any connected graph, where D-sum graph is obtained by using certain D operations on a connected graph. The obtained results are general extensions of the results of Deng et al. [Applied Mathematics and Computation 275(2016): 422-431] and Akhter et al. [AKCE Int. J. Graphs Combin. 14(2017): 70-79] who proved only for γ = 2 and γ = 3, respectively.