Graph invariants in the spin model

Given a symmetric n × n matrix A, we define, for any graph G, f A ( G ) : = ∑ ϕ : V G → { 1 , … , n } ∏ u v ∈ E G a ϕ ( u ) , ϕ ( v ) . We characterize for which graph parameters f there is a complex matrix A with f = f A , and similarly for real A. We show that f A uniquely determines A, up to permuting rows and (simultaneously) columns. The proofs are based on the Nullstellensatz and some elementary invariant-theoretic techniques.