Powers of edge ideals of weighted oriented graphs with linear resolutions.

Let = (V () , E ()) be a weighted oriented graph and I () denote the corresponding edge ideal. In this paper, we give a combinatorial characterization of I () which has a linear resolution. As a consequence, we prove that if I () is the edge ideal of a weighted oriented graph , then I () has a linear resolution if and only if all powers of I () have a linear resolution. Also, we prove that if is a weighted oriented graph and w (x) > 1 for all x ∈ V () , then I () has a linear resolution if and only if all powers of I () 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. [ABSTRACT FROM AUTHOR]