Positivstellensätze for quantum multigraphs

Studying inequalities between subgraph- or homomorphism-densities is an important topic in graph theory. Sums of squares techniques have proven useful in dealing with such questions. Using an approach from real algebraic geometry, we strengthen a Positivstellensatz for simple quantum graphs by Lovász and Szegedy, and we prove several new Positivstellensätze for nonnegativity of quantum multigraphs. We provide new examples and counterexamples.