Depth and Stanley depth of the edge ideals of square paths and square cycles

摘要

In this paper, we compute depth and Stanley depth for the quotient ring of the edge ideal associated to a square path on n vertices. We also compute depth and Stanley depth for the quotient ring of the edge ideal associated to a square cycle on n vertices, when n (math) 0, 3, 4(mod 5), and give tight bounds when n (math) 1, 2(mod 5). We also prove a conjecture of Herzog presented in [5], for the edge ideals of square paths and square cycles.