# Is this rule for graph isomorphism true?

Is it true that two graphs are isomorphic if they:

• Have the same number of vertices;
• Have the same degree for each vertex, that is a graph with degrees $(2,3,2,3)$ would be the same as a graph with degrees $(2,2,3,3)$.
• Have no loops;
• Are connected.

I've made a few tests and this seems to be true. But I don't know how to prove it and hence am not completely sure about it.

• no: o-o+o-o o-o-o+o – John Dvorak Sep 28 '14 at 22:43
• @JanDvorak What is this notation? I presume that the o is a vertex, but am in doubt about - and +. – Billy Rubina Sep 28 '14 at 22:48
• - are edges, and + represents three vertices and four edges. Unfortunately, it's hard to draw a branching structure in ASCII – John Dvorak Sep 28 '14 at 22:58
• another example, from chemistry: 2-methyl-pentene (isohexane) and 3-methyl-pentene (with the methyl group sitting in the middle) – John Dvorak Sep 28 '14 at 22:59