Abstract
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 $R=k[x_1,\ldots, x_n]$.
Highlights
An oriented graph is an ordered pair D = (V (D), E(D)) with the vertex set V (D), the edge set E(D) and an underlying graph G on which each edge is given an orientation
We focus on the Castelnuovo-Mumford regularity, projective dimension and the extremal Betti numbers of M = R/I where I is a homogeneous ideal of R
In the main result of this section, we show that the weight reduction of a non-trivial sink vertex keeps the projective dimension unchanged
Summary
An oriented graph is an ordered pair D = (V (D), E(D)) with the vertex set V (D), the edge set E(D) and an underlying graph G on which each edge is given an orientation. If e = {x, y} is an edge in G and e is oriented from x to y in D, we denote the oriented edge by (x, y) to reflect the orientation. In contrast to directed graphs, multiple edges or loops are not allowed in oriented graphs. An oriented graph D is called vertex-weighted oriented (or weighted ) if each vertex is assigned a weight by a function w : V (D) → N+ called a weight function. If the weight value wi of vertex xi is one, we say xi has a simple weight in D. We say xi has a non-simple weight in D
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