On powers of edge ideals associated to the cycle graph

Consider R is a polynomial ring of m-variables, i.e., [Formula: see text]. In graph G, each edge [Formula: see text] can be associated with a monomials [Formula: see text]. Every edge ideal [Formula: see text] is a monomial ideal of the second degree, which is derived from monomials that correspond to all edges of G. The strategy of this paper is to study the connection between the cycle graphs and its edge ideals. It involves an in-depth analysis of the generators of different powers within monomial ideals of cycle graph [Formula: see text]. We construct specific conditions on the powers of edge ideals to work on its higher powers. Furthermore, we prove their results in both odd and even length cycle graphs