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Algebraic properties of product of graphs

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.