Enumerative applications of a decomposition for graphs and digraphs

A simple decomposition for graphs yields generating functions for counting graphs by edges and connected components. A change of variables gives a new interpretation to the Tutte polynomial of the complete graph involving inversions of trees. The relation between the Tutte polynomial of the complete graph and the inversion enumerator for trees is generalized to the Tutte polynomial of an arbitrary graph. When applied to digraphs, the decomposition yields formulas for counting digraphs and acyclic digraphs by edges and initially connected components.