Actually, I wouldn't call what IGReorderVertices does relabelling. It really just changes the order in which vertices are stored. This in turn affects how certain functions work: the result of AdjacencyMatrix, the order in which BetweennessCentrality returns results, or what DirectedGraph[graph, "Acyclic"] does. It's more about changing the practical representation of the graph than doing anything of mathematical significance.

For true relabelling, you can use VertexReplace.

Example:

If we have

g = Graph[{1, 2, 3}, {1 <-> 2, 2 <-> 3}]

then

IGReorderVertices[{1, 3, 2}, g] returns the equivalent of

g = Graph[{1, 3, 2}, {1 <-> 2, 2 <-> 3}]

Note that the edge list has not changed. 2 is still the "middle" vertex.

VertexReplace[g, {1 -> 1, 2 -> 3, 3 -> 2}] return the equivalent of

g = Graph[{1, 3, 2}, {1 <-> 3, 3 <-> 2}]

Now the edge list has changed too, as 2 was renamed to 3 and 3 was renamed to 2.

If the relabelling is an isomorphic one, then of course the edge list doesn't change (by definition), except in its ordering. Thus, in this case, either function could be used.

In[19]:= perms = GroupElements@GraphAutomorphismGroup[g]
Out[19]= {Cycles[{}], Cycles[{{1, 3}}]}

We could do

VertexReplace[g, Thread[VertexList[g] -> Permute[VertexList[g], #]]] & /@ perms

or, keeping in mind that these re-labellings are isomorphic, simply use

IGReorderVertices[Permute[VertexList[g], #], g] & /@ perms