Let [Formula: see text] be a weighted oriented graph with the underlying graph [Formula: see text] when vertices with non-trivial weights are sinks and [Formula: see text] be the edge ideals corresponding to [Formula: see text] and [Formula: see text] respectively. We give an explicit description of the symbolic powers of [Formula: see text] using the concept of strong vertex covers. We show that the ordinary and symbolic powers of [Formula: see text] and [Formula: see text] behave in a similar way. We provide a description for symbolic powers and Waldschmidt constant of [Formula: see text] for certain classes of weighted oriented graphs. When [Formula: see text] is a weighted oriented odd cycle, we compute [Formula: see text] and prove [Formula: see text] and show that equality holds when there is only one vertex with non-trivial weight.

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Let D be a weighted oriented graph with the underlying graph G when vertices with non-trivial weights are sinks and I(D), I(G) be the edge ideals corresponding to D and G, respectively. We give explicit description of the symbolic powers of I(D) using the concept of strong vertex covers. We show that the ordinary and symbolic powers of I(D) and I(G) behave in a similar way. We provide a description for symbolic powers and Waldschmidt constant of I(D) for certain classes of weighted oriented graphs. When D is a weighted oriented odd cycle we compute reg(I(D) (s) I(D) s ) and prove reg I(D) (s) ≤ reg I(D) s and show that equality holds when there is only one vertex with non-trivial weight.

Let D be a weighted oriented graph and I(D) be its edge ideal. We provide one method to find all the minimal generators of I ⊆C , where C is a maximal strong vertex cover of D and I ⊆C is the intersections of irreducible ideals associated to the strong vertex covers contained in C. If D ′ is an induced digraph of D, under certain condition on the strong vertex covers of D ′ and D, we show that I(D ′ ) (s) ≠ I(D ′ ) s for some s ≥ 2 implies I(D) (s) ≠ I(D) s . We characterize all the maximal strong vertex covers of D such that at most one edge is oriented into each of its vertex and w(x) ≥ 2 if deg D (x) ≥ 2 for all x ∈ V (D). If D is a weighted rooted tree with degree of root is 1 and w(x) ≥ 2 when deg D (x) ≥ 2 for all x ∈ V (D), we show that I(D) (s) = I(D) s for all s ≥ 2.