Distinguishing graphs by their left and right homomorphism profiles

We introduce a new property of graphs called ‘qq-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 HH homomorphism functions from multigraphs GG to HH are specializations of the Tutte polynomial of GG, in particular answering a question of Freedman, Lovász and Schrijver. We also determine for which edge-weighted graphs HH homomorphism functions from multigraphs GG to HH 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.