Abstract. The enumeration of forests of different orders in a graph is carried out by a procedure that involves formal sums and certain annihilation operators on the space of such sums. The results here extend the well-known matrix-tree theorem to the general case of forests.Introduction. In pursuing the enumeration of connected spanning subgraphs [1], each containing a fixed number of cycles, for a planar graph, we introduced a formal procedure together with "annihilation" operators acting on face matrices. From the viewpoint of graphic duality, the face matrix is just the Kirchhoff matrix of the dual graph [2]. In other words, a spanning subgraph with m independent cycles corresponds to a (m + l)-forest in the dual graph. It is therefore obvious that our previous result naturally applies to the counting of spanning forests for a planar graph. In this article, by a further generalization, the use of the formal procedure via the annihilation operators1 enables us to solve the enumeration problems for nonplanar graphs. The method provides again, as in the planar case, some slick expressions for the various sums appearing in the enumeration formulae. Especially in the case of spanning forests of all possible component orders, this approach leads to an expression which is simply the Kirchhoff matrix acted upon by an exponentiation of the annihilation operators we introduced.It is a pleasure to thank the referee for many valuable suggestions which resulted in a substantial improvement of the manuscript.

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