| Summary: | Mathematical modeling or numerical coding of the molecular structures play a significant role in the studies of the quantitative structure-activity relationships (QSAR) and quantitative structures property relationships (QSPR). In 1972, the entire energy of <inline-formula> <tex-math notation="LaTeX">$\pi $ </tex-math></inline-formula>-electron of a molecular graph is computed by the addition of square of degrees (valencies) of its vertices (nodes). Later on, this computational result was called by the first Zagreb index and became well studied topological index in the field of molecular graph theory. In this paper, for <inline-formula> <tex-math notation="LaTeX">$k\in N$ </tex-math></inline-formula> (set of counting numbers), we define four subdivision-related operations of graphs in their generalized form named by <inline-formula> <tex-math notation="LaTeX">$S_{k}$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$R_{k}$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$Q_{k}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$T_{k}$ </tex-math></inline-formula>. Moreover, using these operations and the concept of the cartesian product of graphs, we construct the generalized <inline-formula> <tex-math notation="LaTeX">$F_{k}$ </tex-math></inline-formula>-sum graphs <inline-formula> <tex-math notation="LaTeX">$\Gamma _{1 +F_{k}} \Gamma _{2}$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$F_{k}\in \{S_{k}, R_{k}, Q_{k}, T_{k}\}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\Gamma _{i}$ </tex-math></inline-formula> are any connected graphs for <inline-formula> <tex-math notation="LaTeX">$i\in \{1,2\}$ </tex-math></inline-formula>. Finally, the first and second Zagreb indices are computed for the generalized <inline-formula> <tex-math notation="LaTeX">$F_{k}$ </tex-math></inline-formula>-sum graphs in terms of their factor graphs. In fact, the obtained results are a general extension of the results Eliasi <italic>et al.</italic> and Deng <italic>et al.</italic> who studied these operations for exactly <inline-formula> <tex-math notation="LaTeX">$k=1$ </tex-math></inline-formula> and computed the Zagreb indices for only <inline-formula> <tex-math notation="LaTeX">$F_{1}$ </tex-math></inline-formula>-sum graphs respectively.
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