Homomorphisms and Polynomial Invariants of Graphs 1

This paper initiates a study of the connection between graph homomorphisms and the Tutte polynomial. This connection enables us to extend the study to other important polynomial invariants associated with graphs, and closely related to the Tutte polynomial. We then obtain applications of these relationships in several areas, including Abelian Groups and Statistical Physics. A new type of uniqueness of graphs, strongly related to chromatically-unique graphs and Tutte-unique graphs, is introduced in order to provide a new point of view of the conjectures about uniqueness of graphs stated by Bollobas, Peabody and Riordan.