How many isomorphic graphs does this iso class of 5 vertices and 5 edges have?

Question

I am referring to the second graph. It has 60 graphs, but I can't seem to understand why. What i have so far is that there are (5 choose 2) ways of picking b and d combo; but what do I times this by to get 60? Thank you!

Answer 1

To get the number of graphs for the second graph, we start by choosing a triangle. There are $\binom{5}{3} = 10$ ways to do this. We then choose two of those vertices for to be adjacent to $c$ and $e$. There are $\binom{3}{2} = 3$ ways to do this. Then we choose $c$ and $e$, and there are two ways to do this. So by rule of product, $10 * 3 * 2 = 60$.