Dual graph homomorphism functions

For any two graphs F and G, let hom ( F , G ) denote the number of homomorphisms F → G , that is, adjacency preserving maps V ( F ) → V ( G ) (graphs may have loops but no multiple edges). We characterize graph parameters f for which there exists a graph F such that f ( G ) = hom ( F , G ) for each graph G. sult may be considered as a certain dual of a characterization of graph parameters of the form hom ( . , H ) , given by Freedman, Lovász and Schrijver [M. Freedman, L. Lovász, A. Schrijver, Reflection positivity, rank connectivity, and homomorphisms of graphs, J. Amer. Math. Soc. 20 (2007) 37–51]. The conditions amount to the multiplicativity of f and to the positive semidefiniteness of certain matrices N ( f , k ) .