A polynomial dual of partitions

Let P(x) = Σ_i=0ⁿa_ixⁱ be a nonnegative integral polynomial. The polynomial P(x) is m-graphical, and a multi-graph G a realization of P(x), provided there exists a multi-graph G containing exactly P(1) points where a_i of these points have degree i for 0≤i≤n. For multigraphs G, H having polynomials P(x), Q(x) and number-theoretic partitions (degree sequences) π, ?, the usual product P(x)Q(x) is shown to be the polynomial of the Cartesian product G × H, thus inducing a natural product π? which extends that of juxtaposing integral multiple copies of ?. Skeletal results are given on synthesizing a multi-graph G via a natural Cartesian product G₁ × … × G_k having the same polynomial (partition) as G. Other results include an elementary sufficient condition for arbitrary nonnegative integral polynomials to be graphical.