Regularity in weighted oriented graphs

Let $D$ be a weighted oriented graph with the underlying graph $G$ and $I(D), I(G)$ be the edge ideals corresponding to $D$ and $G$ respectively. We show that the regularity of edge ideal of a certain class of weighted oriented graph remains same even after adding certain kind of new edges to it. We also establish the relationship between the regularity of edge ideal of weighted oriented path and cycle with the regularity of edge ideal of their underlying graph when vertices of $V^+$ are sinks.