Number of Connected simple graphs with n vertices

Question

Is there a relation between numbers of simple graphs and simple connected graphs on n vertices? After observing n=1,2,3,4...., I noticed that the number of connected graphs is at least half of the total number of simple graphs. How can we prove that?

Answer 1

For the part where you asked for Number of Simple Graphs with N vertices I will refer you to a proof in this link as it has been answered here.

Show that the number of simple graphs with $n$ vertices is $2^{{n}\choose{2}}$

Answer 2

The number of nonisomorphic simple graphs is A88 and the number of connected simple graphs is A1349 in the OEIS. They are related by the Euler transform; this is a special case of the Multiset Transformation, which means the not necessarily connected graphs are multisets of the connected graphs.