  Given a vertex-weighted oriented graph, we can associate to it a set of
monomials. We consider the toric ideal whose defining map is given by these
monomials. We find a generating set for the toric ideal for certain classes of
graphs which depends on the combinatorial structure and weights of the graph.
We provide a result which is analogous to the unweighted, unoriented graph
case, to show that when the associated simple graph has only trivial even
closed walks, the toric ideal is the zero ideal. Moreover, we give necessary
and sufficient conditions for the toric ideal of a weighted oriented graph to
be generated by a single binomial and we describe the binomial in term of the
structure of the graph.
