Algebraic invariants of weighted oriented graphs

Abstract

Let $$\mathcal {D}$$ be a weighted oriented graph and let $$I(\mathcal {D})$$ be its edge ideal in a polynomial ring R. We give the formula of Castelnuovo–Mumford regularity of $$R/I(\mathcal {D})$$ when $$\mathcal {D}$$ is a weighted oriented path or cycle such that edges of $$\mathcal {D}$$ are oriented in one direction. Additionally, we compute the projective dimension for this class of graphs.