Dual graph homomorphism functions

Abstract

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. The result 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). © 2009 Elsevier Inc. All rights reserved.