  Let $G$ and $H$ be two simple graphs and let $G*H$ denotes the graph
theoretical product of $G$ by $H$. In this paper we provide some results on
graded Betti numbers, Castelnuovo-Mumford regularity, projective dimension,
$h$-vector, and Hilbert series of $G*H$ in terms of that information of $G$ and
$H$. To do this, we will provide explicit formulae to compute graded Betti
numbers, $h$-vector, and Hilbert series of disjoint union of complexes. Also we
will prove that the family of graphs whose regularity equal the maximum number
of pairwise $3$-disjoint edges, is closed under product of graphs.
