Generating function for representations of graphs by $k$-partite graphs

A relation between the generating function of $k$-partite graphs and the generating function of the number of $k$-partite representations of graphs is obtained. A correlation between the relation's coefficients and chromatic polynomial coefficients is shown. An application of the results to calculation of weighted sums is demonstrated. Special cases of sums and some applications of the relations in physics and mathematics are considered.