On the linear strand of edge ideals of some zero-divisor graphs

Abstract

Let I(G) be the edge ideal associated to a simple graph G. Given a prime number p and an integer, we study the -graded Betti numbers that appear in the linear strand of the minimal free resolution of , where is the zero-divisor graph of the ring . In particular, we compute the extremal Betti number of . As a consequence, we find the Castelnuovo-Mumford regularity and projective dimension of these ideals. Moreover, we exhibit formulae that determine all -graded Betti numbers in the linear strand of the minimal free resolution of for certain values of n.