Number of isomorphisms between two graphs

Question

I'm studying for an exam in graph theory, and this question came up. The question is: how many isomorphisms exist between these two graphs. I know that, as they are isomorphic, this is the same as answering how many isomorphisms there are from the graph to itself.

I think the answer is $6$ or $3$, but I'm struggling to prove/show why this is correct/incorrect. Is there an easy way to see what the correct answer is, without explicitly writing down all the possible bijections?

Thanks

edit: I've included the graphs as a link now.

Answer 1

Consider an arbitrary isomorphism, $G'$, of $G$, and its corresponding bijection, $f$. We must preserve the adjacency and non adjacency of $G$.

Let us start (without loss of generalization) with $a$, which we will map to an arbitrary vertex of degree 2 in $G'$. We have 4 choices possible. Notice that no matter which vertex you pick, its neighbors will be $v$, a vertex of degree 2, and $w$, a vertex of degree 3. Since we must preserve the adjacency and non adjacency of $G$, it follows that the vertex of degree 3 must be labelled $b$, and the vertex of degree 2 must be labelled $f$.

It should be very easy to complete the proof after this point, since our first choice forces every other vertex to be a certain label.

I hope this helped!