What's the equivalent of the adjacency relation for a directed graph? I've found several sources describing a relation notated $\sim$ signifying adjacency in an undirected graph, but nothing explicitly describing an equivalent for a directed graph. I've been using $\overset{\mathit{member}}{\longrightarrow}$, mainly because I have different types of edges so I need a long symbol to overset with the type of the edge linking the nodes. Is this appropriate? Is there a more usual notation?

An arrow is absolutely appropriate to use in this case. For example, see Shimon Even's _Graph Algorithms_. In fact, I think it would be questionable to use anything _but_ an arrow for a directed edge.

Note that Even places the name of the edge over the arrow; your notation of putting the edge type over the arrow is fine too. Or you might put the type underneath, to save room above for the name in case you might need it later.