Betti numbers for the edge ideals of chain graphs and cozero divisor graphs

Let [Formula: see text] be the edge ideal of a chain graph [Formula: see text] with [Formula: see text] vertices. We show that [Formula: see text] is [Formula: see text]-linear and compute the projective dimension and the Betti numbers of [Formula: see text], where [Formula: see text]. As an application, we consider the cozero divisor graph [Formula: see text] of the integers modulo [Formula: see text], and find the aforementioned homological invariants for [Formula: see text] whenever [Formula: see text] gives rise to a chain graph or is a product of three pairwise distinct primes.