[gilleain]

Isomorphism of a pair of graphs usually refers to isomorphism of their unlabelled equivalents.

Yes the concrete expression of the isomorphism would be as a mapping between the labels.

Given that the paper linked to is by Brendan McKay et al, it seems reasonable to mention that nAUTy works by finding (efficiently) all permutations of the labellings that result in an automorphism of the graph.