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

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