Graph invariants in the spin model

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