Regularity of symbolic powers of edge ideals of unicyclic graphs

TL;DR

This paper proves that for unicyclic graphs, the regularity of the s-th symbolic power of the edge ideal equals that of its s-th ordinary power, establishing a key algebraic property.

Contribution

It demonstrates the equality of regularity between symbolic and ordinary powers of edge ideals specifically for unicyclic graphs, a new result in combinatorial commutative algebra.

Findings

Regularity of symbolic and ordinary powers are equal for unicyclic graphs.

The result holds for all positive integers s.

Provides insight into the algebraic structure of edge ideals.

Abstract

Let $G$ be a unicyclic graph with edge ideal $I(G)$. For any integer $s\geq 1$, we denote the $s$-th symbolic power of $I(G)$ by $I(G)^{(s)}$. It is shown that ${\rm reg}(I(G)^{(s)})={\rm reg}(I(G)^s)$, for every $s\geq 1$.