State of The Art
The best current theoretical algorithm is due to Eugene Luks (1983), and is based on the earlier work by Luks (1981), Babai & Luks (1982), combined with a subfactorial algorithm due to Zemlyachenko (1982). The algorithm relies on the classification of finite simple groups. Without CFSG, a slightly weaker bound 2O(√n log2 n) was obtained first for strongly regular graphs by László Babai (1980), and then extended to general graphs by Babai & Luks (1982). Improvement of the exponent √n is a major open problem; for strongly regular graphs this was done by Spielman (1996). For hypergraphs of bounded rank, a subexponential upper bound matching the case of graphs, was recently obtained by Babai & Codenotti (2008).
On a side note, the graph isomorphism problem is computationally equivalent to the problem of computing the automorphism group of a graph, and is weaker than the permutation group isomorphism problem, and the permutation group intersection problem. For the latter two problems, Babai, Kantor and Luks (1983) obtained complexity bounds similar to that for the graph isomorphism.
There are several competing practical algorithms for graph isomorphism, due to McKay (1981), Schmidt & Druffel (1976), Ullman (1976), etc. While they seem to perform well on random graphs, a major drawback of these algorithms is their exponential time performance in the worst case.
Read more about this topic: Graph Isomorphism Problem
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