Regularity of powers of edge ideal of whiskered cycles

TL;DR

This paper establishes an exact formula for the regularity of powers of the edge ideal of whiskered cycle graphs, revealing a precise linear growth pattern related to the cycle length.

Contribution

It provides a closed-form expression for the regularity of all powers of the edge ideal of whiskered cycles, a result previously unknown.

Findings

The regularity of I^s is exactly 2s plus a cycle-dependent constant.

The formula holds for all s ≥ 1.

The result advances understanding of algebraic properties of whiskered cycle graphs.

Abstract

Let $G=W(C_{n})$ be a whiskered cycle graph with edge ideal $I=I(G)$. We prove that for every $s\geq 1$, the equality ${\rm reg}(I^{s})=2s+\lceil \frac{n-1}{2}\rceil-1$ holds.