Abstract:

Let [see pdf for notation] be a nonnegative integral polynomial.
The polynomial P(x) is mgraphical, and a multigraph G a realization of P(x), provided there exists a multigraph G containing exactly P(1) points where [see pdf for notation] of these points have degree i for [see pdf for notation]. For multigraphs G,H having polynomials P(x), Q(x) and numbertheoretic partitions (degree sequences) [see pdf for notation], the usual product P(x)Q(x) is shown to be the polynomial of the Cartesian product [see pdf for notation], thus inducing a natural product [see pdf for notation] which extends that of juxtaposing integral multiple copies of [see pdf for notation]. Skeletal results are given on synthesizing a multigraph G via a natural Cartesian product [see pdf for notation] having the same polynomial (partition) as G. Other results include an elementary sufficient condition for arbitrary nonnegative integral polynomials to be graphical. 