The sequentially Cohen-Macaulay property of edge ideals of edge-weighted graphs

TL;DR

This paper characterizes when the edge ideal of an edge-weighted graph is sequentially Cohen-Macaulay, showing it holds for all weights if and only if the underlying graph is a Woodroofe graph.

Contribution

It provides a complete characterization of the class of graphs whose weighted edge ideals are sequentially Cohen-Macaulay for all weight functions.

Findings

Edge ideal is sequentially Cohen-Macaulay iff the graph is a Woodroofe graph.

Characterization holds universally for all weight functions.

Connects graph structure with algebraic properties of weighted edge ideals.

Abstract

Let $I(G_{\mathbf{w}})$ be the edge ideal of an edge-weighted graph $(G,\mathbf{w})$. We prove that $I(G_{\mathbf{w}})$ is sequentially Cohen-Macaulay for all weight functions $w$ if and only if $G$ is a Woodroofe graph.