Graph Polynomials: From Recursive Definitions To Subset Expansion   Formulas
  Many graph polynomials, such as the Tutte polynomial, the interlace polynomial and the matching polynomial, have both a recursive definition and a defining subset expansion formula. In this paper we present a general, logic-based framework which gives a precise meaning to recursive definitions of graph polynomials. We then prove that in this framework every recursive definition of a graph polynomial can be converted into a subset expansion formula. 
Benny Godlin, Emilia Katz, Johann A. Makowsky