The exponential generating functions for sequence of the numbers of $k$-partite graphs

A specific kind of exponential generating functions for the sequence of the numbers of $k$-partite graphs is considered. These functions take into account the numbers of vertices in each part. A relation is obtained for such generating functions. This relation is a variant of the exponential theorem for these generating functions. It is concluded that it is possible to generalize the obtained relation for hypergraphs and multigraphs. The obtained expression and its simplified special cases are analyzed. The applications of the relations and special cases in physics and mathematics are considered.