The enumeration of bipartite graphs

The standard construction of graphs with n connected components is modified here for bicolored graphs by letting S_n × H act on the function space $\text{Y}{}^1\text{where}\text{X={1,2,…,n}, Y}$ is the set of connected bicolored graphs, and H is the group that interchanges the vertex colors. Then DeBruijn's Generalization of Polya's Theorem is applied to arrive at a direct algebraic relationship between the generating functions for bicolored and connected bicolored graphs. As the former generating function is easily computable, this relationship gives us the latter generating function which is precisely the generating function for connected bipartite graphs.