Let us call graphs $G = (V,E)$ and $G' = (V', E')$ fundamentally different if they are not isomorphic. How many fundamentally different graphs are there on four vertices?
This is a question on my homework. I'm thinking that I need to exhaust all the possible variations of a graph with four vertices:
Each vertices could have a degree of 0, 1, 2 or 3.
Four possibilities times 4 vertices = 16 possibilities.
And also, maybe, since the graphs are fundamentally different (not isomorphic), you need to minus 1 possible variation since it would match the other graph.