Algebraic Properties of Powers of Edge Ideals of Vertex-Weighted Oriented Cycles

Abstract

Let [Formula: see text] be any positive integer and [Formula: see text] the edge ideal of a vertex-weighted oriented [Formula: see text]-cycle graph [Formula: see text]. We provide explicit formulas for the regularity and depth of [Formula: see text]. In particular, we find that the regularity of [Formula: see text] is a linear function; [Formula: see text], the set of associated prime ideals of [Formula: see text], equals [Formula: see text]; and the depth of [Formula: see text] is a constant for all [Formula: see text]. We also give some examples to show that these results are related to the direction selection of edges and the weight of vertices.