Graph isomorphism problem is one of the longest standing problems that resisted classification into $P$ or $NP$-complete problems. We have evidences that it can not be $NP$-complete. Firstly, Graph Isomorphism can not be $NP$-complete unless the polynomial hierarchy [1] collapses to the second level. Also, the counting[2] version of GI is polynomial-time Turing equivalent to its decision version which does not hold for any known $NP$-complete problem. The counting version of *natural* $NP$-complete problems seems to have much higher complexity since they are $\#P$-complete. Finally, the lowness result [3] of GI with respect to $PP$ ($PP^{GI}=PP$) is not known to hold for any $NP$-complete problem. The lowness result of GI has been improved to $SPP^{GI}=SPP$ after Arvind and Kurur proved that GI is in $SPP$ [4].


**Edit**: It is known that SAT is truth-table equivalent to USAT (set of Boolean formulas with exactly one satisfying assignment) while it is not known whether GI and UGI are equivalent under polynomial time truth-table reductions.



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> What other (recent) results can provide further evidence that GI can not be $NP$-complete?
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[1]: Uwe Schöning, "Graph isomorphism is in the low hierarchy", Proceedings of the 4th Annual Symposium on Theoretical Aspects of Computer Science, 1987, 114–124


[2]: R. Mathon, "A note on the graph isomorphism counting problem", Information Processing Letters, 8 (1979) pp. 131–132


[3]: Köbler, Johannes; Schöning, Uwe; Torán, Jacobo (1992), "Graph isomorphism is low for PP", Computational Complexity 2 (4): 301–330


[4]: V. Arvind and P. Kurur. Graph isomorphism is in SPP, ECCC TR02-037, 2002.



Evidence that Graph Isomorphism problem is not $NP$-complete
A search done for a [related question](https://cstheory.stackexchange.com/questions/32160/what-evidence-is-there-that-graph-isomorphism-is-not-in-p) also turned up (as indicated in a comment) [rjlipton.wordpress.com/2015/03/05/news-on-intermediate-problems](https://rjlipton.wordpress.com/2015/03/05/news-on-intermediate-problems/):



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> Now Eric and Bireswar [complete](http://eccc.hpi-web.de/report/2014/068/) the triad of relations to the other intermediate problems:
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> > **Theorem 2** *Graph Isomorphism is in ${\mathsf{RP}^{\mathrm{MCSP}}}$. Moreover, every promise problem in ${\mathsf{SZK}}$ belongs to ${\mathsf{BPP}^{\mathrm{MCSP}}}$ as defined for promise problems.*
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Here $\mathsf{RP}$ stands for [randomized polynomial time](https://en.wikipedia.org/wiki/RP_(complexity)), MCSP is the Minimum Circuit Size Problem, and $\mathsf{SZK}$ stands for Statistical Zero Knowledge. So we learn that if MCSP is not NP-complete and RP!=NP, then graph isomorphism is not NP-complete either.


