Powers of edge ideals of weighted oriented graphs with linear resolutions

Abstract

Let [Formula: see text] be a weighted oriented graph and [Formula: see text] denote the corresponding edge ideal. In this paper, we give a combinatorial characterization of [Formula: see text] which has a linear resolution. As a consequence, we prove that if [Formula: see text] is the edge ideal of a weighted oriented graph [Formula: see text], then [Formula: see text] has a linear resolution if and only if all powers of [Formula: see text] have a linear resolution. Also, we prove that if [Formula: see text] is a weighted oriented graph and [Formula: see text] for all [Formula: see text], then [Formula: see text] has a linear resolution if and only if all powers of [Formula: see text] have linear quotients. We provide a lower bound for the regularity of powers of edge ideals of weighted oriented graphs in terms of induced matching. Finally, we obtain a general upper bound for the regularity of edge ideals of weighted oriented graphs.