Projective dimension and regularity of edge ideals of some vertex-weighted oriented unicyclic graphs

In this paper we provide some exact formulas for the projective dimension and the regularity of edge ideals of vertex-weighted oriented unicyclic graphs. These formulas are in function of the weight of the vertices, the numbers of edges. We also give some examples to show that these formulas are related to direction selection and the assumptions that $w(x)\ge 2$ for any vertex $x$ cannot be dropped.