Abstract
We introduce a new property of graphs called ''q-state Potts uniqueness and relate it to chromatic and Tutte uniqueness, and also to ''chromatic-Flow uniqueness'', recently studied by Duan. Wu and Yu.
We establish for which edge-weighted graphs H homomorphism functions from multigraphs G to H are specializations of the Tutte polynomial of G, in particular answering a question of Freedman, Lovasz and Schrijver. We also determine for which edge-weighted graphs H homomorphism functions from multigraphs G to H are specializations of the ''edge elimination polynomial'' of Averbouch, Godlin and Makowsky and the ''induced subgraph polynomial'' of Tittmann, Averbouch and Makowsky.
Unifying the study of these and related problems is the notion of the left and right homomorphism profiles of a graph.