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!

For labeled vertices:
To count undirected loopless graphs with no repeated edges, first count possible edges. As Andre counts, there are $\binom{n}{2}$ such edges. One by one, each edge is either included or excluded. So this gives $2^{\binom{n}{2}}$ possible graphs.
If loopless graphs with no repeated edges are directed, each pair of vertices $a<b$ provides $3$ possibilities for a (potentially absent) edge. Do you see what they are and how that modifies the count?
If there are loops (but still no repeated edges), then either of the above scenarios are modified by realizing that there are $n$ more pairs of points to consider - the ones where $a=b$. Do you see how this would modify the count of graphs? Watch out - it doesn't really make sense to count a loop as directed.
