Graph theory served in different fields of sciences, especially in chemistry in which creating complex structures and studying their enormous properties. Graph operation is a tool to construct complex chemical structures using basic graphs. While studying their properties, topological descriptors are a well-known methodology introduced by chemists, and even after half of a century past, it is still serving. Formally, a topological descriptor or index is a numerical value corresponding to a chemical structure. This numerical value can be easily accessed by a particular equation, for example, the second Zagreb index, the first reformulated Zagreb, and also from the forgotten topological descriptor. In this particular work, we generalized two existing graph operations, and by using these newly developed graph operations, we created two complex structures by using two graph operations, namely, the corona product and double graph operation. Furthermore, to evaluate the chemical properties of these newly generated structures, we used the methodology of topological descriptors, particularly the first and second Zagreb, the first reformulated Zagreb, and forgotten topological descriptors. Moreover, we also presented the closed formulas of the first and second Zagreb co-indices for these newly generated structures.