TITLE: How many different possible simply graphs are there with vertex set V of n elements 
QUESTION [8 upvotes]: Is V is a set with n elements, how many different simple, undirected graphs are there with vertex set V?

REPLY [8 votes]: See http://oeis.org/A000088. This is the sequence which gives the number of isomorphism classes of simple graphs on n vertices, also called the number of graphs on n unlabeled nodes.  You will also find a lot of relevant references here.