Let us newly define the Subdivision vertex corona and subdivision vertex neighbourhood corona product of the Cycle graph $ C_m $, for any $ m\ge3 $ with empty graph $ \overline{K_n} $, for any $ n \ge1 $. Additionally we will give generalization result based on the graph $ S({C_m}\odot\overline{K_n})$ and $ S_N({C_m}\boxdot\overline{K_n})$ for any, $ m\ge 3, n\ge 1 $. Then we shall calculate the topological indices of the different types of Zagreb index for our newly defined graphs. The molecular graph of Subdivision vertex corona product of $ C_3 $ and $ \overline{K_2} $ i.e., $ S( C_3 \odot \overline{K_2} )$ can be represented in the form of tri-thioacetane.