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

TL;DR

This paper investigates the regularity of symbolic and ordinary powers of edge ideals in weighted oriented graphs, providing explicit formulas, inequalities, and conditions for equality, with a focus on complete graphs and graphs with sink vertices.

Contribution

It introduces new bounds and formulas for the regularity of powers of edge ideals in weighted oriented graphs, extending known results to more general graph classes.

Findings

For complete graphs, regularity of symbolic powers is bounded by that of ordinary powers.

Explicit formulas for regularity of powers in complete graphs are derived.

Regularity of symbolic powers is eventually linear in the power k.

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.