  In this note we study some of the properties of the generating polynomial for
homomorphisms from a graph to at complete weighted graph on $q$ vertices. We
discuss how this polynomial relates to a long list of other well known graph
polynomials and the partition functions for different spin models, many of
which are specialisations of the homomorphism polynomial.
  We also identify the smallest graphs which are not determined by their
homomorphism polynomials for $q=2$ and $q=3$ and compare this with the
corresponding minimal examples for the $U$-polynomial, which generalizes the
well known Tutte-polynomal.

Combinatorics;Statistical Mechanics;Discrete Mathematics