Lovász's theorem. The forward implication is the substantial one: it factors through the invertibility of the homomorphism matrix of a family of representatives of all graphs on at most $n$ vertices, which follows from its factorisation into the surjective-homomorphism matrix, the diagonal of automorphism counts, and the injective-homomorphism matrix, both outer factors being triangular with positive diagonal. The converse is the invariance of homomorphism counts under isomorphism of the target.