How many isomorphisms are there of unlabeled 2 vertex graphs, and how many of labeled 2 vertex graphs? Loops are allowed.
I know this is trivial but I suspect there're 4 unlabeled, no loops, one edge; no edge, two loops; one loop, one edge; one loop no edge; Would labeling halve this?
Labeling refers to the vertices.
The number of loops can be zero, one, or two; that's $3$ possibilities.
The number of edges can be zero, or one; that's $2$ possibilities.
And each of the first $3$ possibilities can be combined with each of the last $2$ possibilities, making $3 \times 2 = 6$ different isomorphism types in total. So, you forgot some.