From the Ising and Potts models to the general graph homomorphism polynomial

A graph homomorphism from a graph G to a graph H is a mapping h : V (G)→ V (H) such that h(u) ∼ h(v) if u ∼ v. Graph homomorphisms are well studied objects and, for suitable choices of eitherG orH, many classical graph properties can be formulated in terms of homomorphisms. For example the question of wether there exists a homomorphism from G to H = Kq is the same as asking wether G is q-colourable or not. A number of classical models in statistical physics, such as the Ising model, Potts model and lattice gas, can be formulated in terms of the generating function for weighted versions of homomorphisms from G to some graph H. We refer the reader to [HN04] for a comprehensive survey of the algebraic aspects of graph homomorphisms. Our aim here is to discuss the generating polynomial for homomorphisms from a graph G to the most general weighted graph on q vertices. For a fixed q this is an object of polynomial size which contains a wealth of informations about the graph G, but as we will later show it is not a complete graph invariant. We will first defined this generating function as a polynomial, then recall the definitions of a number of well known graph polynomials, and partition functions from physics, and then proceed to study the properties and relationships of these polynomials. Let us give the formal definitions of our objects of study.