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 $K_n$, we show that $\text{reg}(I(K_n{)}^{(k)})\le \text{reg}(I(K_n{)}^k)$ for all $k\ge 1$. Also, we give explicit formulas for $\text{reg}(I(K_n{)}^{(k)})$ and $\text{reg}(I(K_n{)}^k)$, for any $k\ge 1$. As a consequence, we show that $\text{reg}(I(K_n{)}^{(k)})$ is eventually a linear function of $k$. For any weighted oriented graph $D$, if $V^{+}$ are sink vertices, then we show that $\text{reg}(I(D{)}^{(k)})\le \text{reg}(I(D{)}^k)$ with $k=2,3$ and equality cases studied. Furthermore, we give formula for $\text{reg}(I(D{)}^2)$ in terms of $\text{reg}(I(D{)}^{(2)})$ and regularity of certain induced subgraphs of $D$. Finally, we compare the regularity of symbolic powers of weighted oriented graphs $D$ and $D^{\mathrm{\prime }}$, where $D^{\mathrm{\prime }}$ is obtained from $D$ by adding a pendant.