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

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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 ( K n ) ( k ) ) ≤ reg ( I ( K n ) k ) for all k ≥ 1 . Also, we give explicit formulas for reg ( I ( K n ) ( k ) ) and reg ( I ( K n ) k ) , for any k ≥ 1 . As a consequence, we show that reg ( I ( K n ) ( 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.