Graphs and algebras of symmetric functions

We describe an algebraic technique for operating with power series whose coefficients are represented by integrals of symmetric functions $f_n$ defined on the Cartesian powers $\Omega^n$ of a set $\Omega$ with a measure $\mu$. Moreover, each of the coefficient functions $f_n$ is obtained by means of a special mapping from graphs with $n$ labeled vertices belonging to a fixed class. This technique has application to equilibrium statistical mechanics and to problems of enumeration of graphs.