How to determine the number of directed/undirected graphs? I'm kind of stuck on this homework problem, could anyone give me a springboard for it? If we have $n\in\mathbb{Z}^+$, and we let the set of vertices $V$ be a set of size $n$, how can we determine the number of directed graphs/undirected graphs/graphs with loops etc.? Is there a formula for this? I feel like it can be done using combinatorics but I can't quite figure it out. Any ideas?
Thanks!

Without some restrictions, there are infinitely many, so I suspect you at least have the assumption that you can't have more than one edge between a single pair of vertices.
You should think about what the maximal number of possible edges you could have is in a graph with $n$ vertices (this is why we should assume no multiple edges, so the number is finite, but it will be different depending on whether you allow loops or not). Then to specify a particular graph on $n$ vertices, you need to specify whether each of the possible edges appears in your particular graph or not. If your graph should be oriented, you then have the additional choice of which way the edge should point. You should then be able to ennumerate the total number of possible choices (if you can't, let me know and I'll add more detail here - as this is tagged homework I don't want to give too much away immediately!).
