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

Abstract: 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 $\reg(I(K_n)^{(k)})\leq \reg(I(K_n)^k)$ for all $k\geq 1$. Also, we give explicit formulas for $\reg(I(K_n)^{(k)})$ and $\reg(I(K_n)^{k})$, for any $k\geq 1$. 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 $\reg(I(D)^{(k)}) \leq \reg(I(D)^k)$ 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.