When are two unlabelled simple graphs considered equal?

Question

When are two unlabelled simple (not necessarily connected) graphs considered equal? I don't really find a way to formally state this.

Additionally, what might be a way to find the number of unlabelled simple graphs (not necessarily connected)? I understand that there isn't a closed form. Is there any recursion or something of that sort which can evaluate this (in some manner)?

Answer 1

For your first question

When are two unlabelled simple (not necessarily connected) graphs considered equal?

to systematically determine when two such graphs are equal (isomorphic) you label both of them and try possible bijections between the nodes. If no such bijection is a graph isomorphism, then the two graphs are not equal.

The graph isomorphism problem is very difficult in general. It is often easier to determine when two graphs are not equal. For example, if they have different degree sequences then they are not equal.

For your second question

Additionally, what might be a way to find the number of unlabelled simple graphs (not necessarily connected)?

the OEIS sequence A000088 is "Number of graphs on n unlabeled nodes" and it has some Mathematica and PARI/GP code that computes the sequence.