A new algorithm of Graph Isomorphism is invented by PCT/CN2020/134861, roughly speaking time complexity $\leqslant5n^2$ except for regular graph, automorphism group can be known as by-product. Peer reviewed paper can be downloaded from www.getpaperfree.com by search "同构", it is written in Chinese.

Except strongly regular graph, all other regular graph's isomorphism can be decided in $O(n^3)$.

After 2 years' hard work, though I am almost sure that time complexity $\lt\lt O(n^{\ln n})$ when $n$ is big enough for all graphs, time complexity $\leqslant O(n^5)$ for almost all strongly regular graphs, , fail to know what's the worst case.

If you know a good classification of strongly regular graphs or other useful algorithm for them, please answer the question. Thank you!

*** update including test code for potential counterexamples, but Sagemath failed to relabel graphs, confused***

import time 
G = graphs.RandomGNP(20,0.4)
CFI, p = graphs.CaiFurerImmermanGraph(G)
H, r =  graphs.CaiFurerImmermanGraph(G,twisted = true)
print(H.degree_sequence() == CFI.degree_sequence())
print(H.automorphism_group()== CFI.automorphism_group())
print(H.automorphism_group().order())
n = H.order()
CFI.relabel(range(n))
Mg = dict()
for i in CFI.vertex_iterator():
    Mg[i] = set(CFI.neighbors(i))
H.relabel(range(n))
Mk = dict()
for i in H.vertex_iterator():
    Mk[i] = set(H.neighbors(i))
print (time.ctime())
print(iso(Mg,Mk,set(range(n)),set(range(n))))
print (time.ctime())
print(H.is_isomorphic(G))
print (time.ctime())