Let I = I(G) be the edge ideal of a graph G. We give various general upper bounds for the regularity function reg I s , for s ≥ 1, addressing a conjecture made by the authors and Alilooee. When G is a gap-free graph and locally of regularity 2, we show that reg I s = 2s for all s ≥ 2. This is a weaker version of a conjecture of Nevo and Peeva. Our method is to investigate the regularity function reg I s , for s ≥ 1, via local information of I.
Let D be a weighted oriented graph and let I(D) be its edge ideal in a polynomial ring R. We give the formula of Castelnuovo-Mumford regularity of R/I(D) when D is a weighted oriented path or cycle such that edges of D are oriented in one direction. Additionally, we compute the projective dimension for this class of graphs.
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 $R=k[x_1,\ldots, x_n]$.
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