Algebraic Properties of Product of Graphs

Abstract

Let G and H be two simple graphs and let G*H denotes the graph theoretical product of G by H. In this article, we provide 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. More generally, we present explicit formulae to compute graded Betti numbers, h-vector, and Hilbert series of disjoint union of complexes. We will also prove that the family of graphs for which the regularity of each edge ring equals the maximum number of pairwise 3-disjoint edges, is closed under product of graphs.