论文标题
Betti数量的加权方向图
Betti numbers of weighted oriented graphs
论文作者
论文摘要
令$ \ mathcal {d} $为加权的图形，$ i（\ mathcal {d}）$为其边缘理想。在本文中，我们通过上Koszul Simplicial Complextes，Betti分割和映射锥构建研究了$ i（\ Mathcal {d}）$的Betti数字。特别是，我们为几类加权图形的边缘理想数量的betti数量提供了递归公式。我们还标识了加权方向的类别的类别，其边缘理想具有独特的极值betti数字，该数字使我们能够计算已识别类的规律性和投影尺寸。此外，我们表征了$ n $ vertices上的加权图形$ \ MATHCAL {d} $的结构，例如$ \ textrm {pdim}（r/i（\ nathcal {d}）= n $ where $ r = k [x_1，\ ldots，x_n] $。
Let $\mathcal{D}$ be a weighted oriented graph and $I(\mathcal{D})$ be its edge ideal. In this paper, we investigate the Betti numbers of $I(\mathcal{D})$ via upper-Koszul simplicial complexes, Betti splittings and the mapping cone construction. In particular, we provide recursive formulas for the Betti numbers of edge ideals of several classes of weighted oriented graphs. We also identify classes of weighted oriented graphs whose edge ideals have a unique extremal Betti number which allows us to compute the regularity and projective dimension for the identified classes. Furthermore, we characterize the structure of a weighted oriented graph $\mathcal{D}$ on $n$ vertices such that $\textrm{pdim } (R/I(\mathcal{D}))=n$ where $R=k[x_1,\ldots, x_n]$.
