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.
Rather et al. (Tue,) studied this question.