The graph theoretic moment problem

We study an analogue of the classical moment problem in the framework where moments are indexed by graphs instead of natural numbers. We study limit objects of graph sequences where edges are labeled by elements of a topological space. Among other things we obtain strengthening and generalizations of the main results of previous papers characterizing reflection positive graph parameters, graph homomorphism numbers, and limits of simple graph sequences. We study a new class of reflection positive partition functions which generalize the node-coloring models (homomorphisms into weighted graphs).