Number of possible graphs of N vertices connected/disconnected.

Question

Motivated by this question here

Number of Connected simple graphs with n vertices,

I referred many webpages to find an apt proof for it But I am now confused because some of the links mention Number of Simple Graphs with $N$ vertices as $2^{\frac{N(N-1)}{2}} $ but For $N=4$ we get 11 graphs in total(Link) and the answer is different for complete series , can anyone Answer it that what is the right answer for number of graphs ?

Some Links mentioned about isomorphic and non -isomorphic graphs , Please explain them too because I didn't get a good resource to read about them.

Answer 1

This is a very common situation in enumeration problems. When we count some mathematical objects do we want to regard isomorphic objects are being essentially the same or not? We say that two graphs that are isomorphic are equal up to isomorphism. This is also known as choosing unlabeled versus labeled objects. Here is a quote from MathWorld:

A graph in which individual nodes have no distinct identifications except through their interconnectivity. Graphs in which labels (which are most commonly numbers) are assigned to nodes are called labeled graphs. Unless indicated otherwise by context, the unmodified term "graph" generally refers to an unlabeled graph.

Thus, in a labeled simple graph with $N$ vertices, each of the $\frac{N(N-1)}2$ edges can be in the graph or not independently. This implies the simple expression $2^{\frac{N(N-1)}2}$ for the number of labeled graphs. For unlabeled graphs the expression is more complicated as in OEIS sequence A000088.