To detect isomorphic graphs Is it enough to check if they have the same number of same degree vertices?

Question

Given two lists of non directional graph edges e.g. [(1,3),(3,5),(5,1),(5,7)] [(4,5),(2,3),(3,4),(4,2)] In order to check if the two graphs are isomorphic is it enough to count the vertices with the same degree between them?

e.g. Vertice   1: 2    1: 0
               2: 0    2: 2
               3: 2    3: 2
             4,6: 0    4: 3
               5: 3    5: 1
               7: 1    6,7: 0

So in our example: both graphs have 2 vertices with 2 edges 1 with 3, one with one and 3 with 0

Answer 1

No. If two graphs have different degree sequences, they are definitely not isomorphic, so the algorithm "half-works". But consider a six cycle versus two three-cycles. Both graphs have six vertices, all of degree two; they are not isomorphic. Indeed, this counterexample shows that nothing based solely on degrees can work.