Is V is a set with n elements, how many different simple, undirected graphs are there with vertex set V?



How many different possible simple graphs are there with vertex set V of n elements
If you consider isomorphic graphs different, then obviously the answer is $2^{n\choose 2}$. Most graphs have no nontrivial automorphisms, so up to isomorphism the number of different graphs is asymptotically $2^{n\choose 2}/n!$. This goes back to a famous method of Pólya (1937), see [this paper](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC528402/) for more information. You can find Pólya's original paper [here](https://doi.org/10.1007/BF02546665).


