Please explain why this isomorphic class has 30 graphs and 4 automorphisms

Question

Please explain why this iso class with 5 vertices and edges has 30 graphs and 4 automorphisms. I understand there are 5 ways to choose a, but then where does 4 choose 2 come in? Please help. This is killing me.

Answer 1

An automorphism is an isomorphism from a graph to itself. There can only be one possible mapping for $a \in V(G)$, and that is to itself. Now, $b, c$ are adjacent to each other. So we can swap their ordering, as they are both distance $1$ from $a$; and they are both distance $2$ from the other graph. Similarly, we can map $d, e$ in two possible ways. So we have:

$a \to a$
$b \to b, c$
$c \to b, c$
$d \to d, e$
$e \to d, e$

And so we have four automorphisms.