Regularity of symbolic and ordinary powers of weighted oriented graphs and their upper bounds

In this paper, we compare the regularities of symbolic and ordinary powers of edge ideals of weighted oriented graphs. For any weighted oriented complete graph Kn, we show that reg(I(Kn)(k))<= reg(I(Kn)k) for all k >= 1. Also, we give explicit formulas for reg(I(Kn)(k)) and reg(I(Kn)k), for any k >= 1. As a consequence, we show that reg(I(Kn)(k)) is eventually a linear function of k. For any weighted oriented graph D, if V+ are sink vertices, then we show that reg(I(D)(k))<= reg(I(D)k) with k = 2, 3 and equality cases studied. Furthermore, we give formula for reg(I(D)2) in terms of reg(I(D)(2)) and regularity of certain induced subgraphs of D. Finally, we compare the regularity of symbolic powers of weighted oriented graphs D and D ', where D ' is obtained from D by adding a pendant.