But it looks the Graph class works on types a and b where a is the Node type where b is the Edge type, or just the other way around :), I don't have the code right now.

Yes, 'Graph n e' is a graph whose nodes are labelled with elements of type n and whose edges are labelled with elements of type e.

The above rules out my way of representing the data, or doesn't it (i.e. is there some fancy, magical way of doing this) ?

Yes. (I.e., it rules it out.)

The second intuitive solution is to take a datatype like data GraphNode|Teacher etc |Subject etc |Student etc

Should this be the way to do it?

Yes.

And then yet another question: I need some kind of way to remember, which possibilities I already had.

In example in my C++ program I used a couple of array's with in each array pointers to the StudentObjects, TeacherObjects.

You could use the same here, or use a FiniteMap. If i'm understanding you correctly you miss the fact that there's no way to look up a node given it's label. This is because the graph module doesn't assume that node labels are unique (otherwise it wouldn't need to generate the Node labels itself). I generally bundle a 'Graph n e' together with a 'FiniteMap n Node' so that I can look up Node identifiers based on node labels.

But then I yet have another problem. How can I guarantee, to do a complete path traversal, each time in exactly the same way?

I don't understand what you mean. In your case, what is a "complete path traversal." Regardless, if you have the same graph and the same functions for traversing it, since Haskell is referentially transparent you should also "traverse it in the same way..." - Hal