Graph-functions associated with an edge-property

Let P be an edge-property of graphs. For any graph G we construct a polynomial Ψ(G, η, P), in an indeterminate η ,i n which the coef fi cient of η r for any r ≥ 0 gives the number of subsets of E(G )t hat have cardinality r and satisfy P. An example is the well known matching polynomial of a graph. After studying the properties of Ψ(G, η, P )i n general, we specialise to two particular edge-properties: that of being an edge-covering and that of inducing an acyclic subgraph. The resulting polynomials, called the edge-cover and acyclic polynomials respectively, are studied and recursive formulae for computing them are derived. As examples we calculate these polynomials for paths and cycles.