In this article, we characterize the solutions of graph equations, \(\varphi (\overline{G})\cong \overline{\varphi (G)}\) (equivalently \(\varphi (G)\cong \overline{\varphi (\overline{G})}\) ) establishing the commutativity of two graph operations; the complement and \(\varphi \) (where \(\varphi \) is one of the graph operations namely; creating of line graph or total graph) on a simple graph G. In order to achieve this aim, estimates on various graph parameters such as order, size, minimum & maximum degree and the sum of the square of the degrees are made. An algebraic approach is developed and applied when the number of graphs with given order and size is large which makes it difficult to deal with them individually. Additionally, it is observed that for a graph G satisfying the aforesaid graph equations, the sum of the degree squared coincides for both G and \(\overline{G}\) .
