  Graph polynomials are graph parameters invariant under graph isomorphisms
which take values in a polynomial ring with a fixed finite number of
indeterminates. We study graph polynomials from a model theoretic point of
view. In this paper we distinguish between the graph theoretic (semantic) and
the algebraic (syntactic) meaning of graph polynomials. We discuss how to
represent and compare graph polynomials by their distinctive power. We
introduce the class of graph polynomials definable using Second Order Logic
which comprises virtually all examples of graph polynomials with a fixed finite
set of indeterminates. Finally we show that the location of zeros and stability
of graph polynomials is not a semantic property. The paper emphasizes a model
theoretic view and gives a unified exposition of classical results in algebraic
combinatorics together with new and some of our previously obtained results
scattered in the graph theoretic literature.

Logic;Combinatorics