Evaluations of Graph Polynomials

A graph polynomial p(G, ¯ X) can code numeric information about the underlying graph G in various ways: as its degree, as one of its specific coefficients or as evaluations at specific points ¯ X = ¯ x0. In this paper we study the question how to prove that a given graph parameter, say !(G), the size of the maximal clique of G, cannot be a fixed coefficient or the evaluation at any point of the Tutte polynomial, the interlace polynomial, or any graph polynomial of some infinite family of graph polynomials. Our result is very general. We give a sufficient condition in terms of the connection matrix of graph parameter f(G) which implies that it cannot be the evaluation of any graph polynomial which is invariantly definable in CMSOL, the Monadic Second Order Logic augmented with modular counting quantifiers. This criterion covers most of the graph polynomials known from the literature.