Evidence that Graph Isomorphism problem is not $NP$-complete
Due to Babai's recent result (see the paper) $GI$ is in quasi-polynomial time ($QP$). If $GI$ is $NP$-complete, then it implies $NP\subseteq  QP=DTIME(n^{polylog\, n})$. This, in turn, implies $EXP=NEXP$, see 
here. Therefore, if the commonly accepted conjecture $EXP\neq NEXP$ holds, then $GI$ cannot be $NP$-complete.