Regularity of powers of edge ideals of Cohen-Macaulay weighted oriented forests

TL;DR

This paper provides explicit combinatorial formulas for the regularity of powers of edge ideals of weighted oriented forests with specific properties, revealing a piecewise linear relationship with the power index.

Contribution

It introduces a novel combinatorial formula for the regularity of powers of edge ideals in weighted oriented forests with leaves as sinks.

Findings

Regularity of edge ideal powers is a piecewise linear function of k.

Formulas are explicitly combinatorial and applicable to Cohen-Macaulay weighted oriented forests.

Results extend understanding of algebraic invariants in combinatorial structures.

Abstract

In this paper, we explicitly give combinatorial formulas for the regularity of powers of edge ideals, $\reg(I(D)^k)$, of weighted oriented unmixed forests $D$ whose leaves are sinks ($V^+(D)$ are sinks). This combinatorial formula is a piecewise linear function of $k$, for $k \geq 1$.