We provide formulas for the projective dimension and the regularity of edge ideals associated to vertex weighted rooted forests and oriented cycles. We derive from that exact formulas for the depth of those ideals. We also give some examples to show that the assumptions cannot be dropped.
"Projective dimension and regularity of edge ideals of some weighted oriented graphs." Rocky Mountain J. Math. 49 (4) 1391 - 1406, 2019. https://doi.org/10.1216/RMJ-2019-49-4-1391