So I said that the answer is yes, because the amount of 3-length cycles in each graph match so the two graphs must be isomorphic.
In order to find an isomorphism I did this.
Let's pick vertex a to correspond with vertex 1, so a-1 (fine via symmetry). a is adjacent to b,e,d like how 1 is adjacent to 2,3,6 so let's assign b-2, e-3, d-6 (also fine via symmetry). So 1, 2, and 6 are adjacent to 4 like how a,b, and d are adjacent to f, so f-4 and that makes c to be assigned to 5 so c-5.
So the graphs are isomorphic. An isomorphism is a-1, b-2, c-5, d-6, e-3, f-4
but the answer in the book says Yes, they are isomorphic. An isomorphism is a-6, b-1, c-3, d-5, e-2, f-4 which does not match my answer.
Where did I go wrong?
Here is the short answer: $a$ can't corresponds to 1. They have different numbers of edges out from the vertex!