TITLE: Are regular graphs the hardest instance for graph isomorphism?
QUESTION [8 upvotes]: Regular graphs are the graphs in which the degree of each vertex is the same. The Weisfeiler-Lehman algorithm fails to distinguish between the given two non-isomorphic regular graphs. 
Is there a fastest known algorithm for regular graph isomorphism? Are regular graphs the hardest instance for graph isomorphism? Is there any combinatorial or algebraic technique (group theoretic) to deal with this situation efficiently?

REPLY [4 votes]: An algorithm which never distinguished between non-isomorphic regular graphs (of the same order)  would be spectacularly weak. Some appropriate canonical form of the adjacency matrix is a (totally unusable) invariant which definitively determines isomorphism.  Any invariant which sometimes fails to distinguish non-isomorphic graphs is likely to have some pairs $G,G'$   which are non-isomorphic, both regular, and not distinguished. 
The article to which you link begins, in part:

The classical Weisfeiler-Lehman method WL[2] uses edge colors to produce a powerful graph invariant. ... Many traditionally used combinatorial invariants are determined by WL[k] for small k. We show that WL[2] determines the number of cycles of lengths up to 6.

So there are some cases of non-isomorphic graphs that WL[2] does not distinguish and certainly some of them are both regular. However such a pair would need to agree on the number of cycles of length $3,4,5,6$ and other things.