Exact Homological Invariants for Closed‑Neighborhood Ideals of Cubes of Broom Graphs

Abstract

Let \(B(n, m)\) be the broom graph formed from a path \(y_{1},\dots, y_{m}\) by attaching n leaves to \(y_{1}\), and let \(I_{n,m}\) be the closed‑neighborhood ideal of its third graph power. Over an arbitrary field, we determine the projective dimension, Castelnuovo–Mumford regularity, and height of \(S / I_{n,m}\) for every \(n,m\ge 2\). The formulas are uniform, including all small values of m. The proof uses minimal multigraded mapping cones and two interval‑ideal recurrences; a disjoint‑support packing and an explicit cover give the height. We also classify the Cohen–Macaulay cases.