Regularity of edge ideals of powers of graphs

TL;DR

The paper investigates the Castelnuovo-Mumford regularity of edge ideals of powers of simple graphs. It proves that for forests the regularity is weakly decreasing along powers, by linking reg to the induced matching number and leveraging chordality of powered forests. For cycles, it derives an exact formula for $\mathrm{reg}(R/I(C_n^d))$, depending on $n$ and $d$, via combinatorial decompositions and induction. These results illuminate how graph structure governs algebraic invariants of powers and provide precise predictions for forests and cycles.

Abstract

We prove that the regularity of edge ideals of powers of forests is weakly decreasing. We then compute the regularity of edge ideals of powers of cycles.

Theorems & Definitions (19)

• Definition 2.1
• proof
• Definition 2.3
• proof
• proof
• proof
• proof : Proof of Theorem \ref{['thm_reg_decrease_forests']}
• Remark 2.7
• Example 2.8
• proof