Is there a simple argument why graph isomorphism is not NP-complete? I need to provide one simple evidence that graph isomorphism (GI) is not NP-complete.
I saw a number of papers on google scholar and answers on StackExchange. However, I have very limited knowledge of graph isomorphism, and I would like to just provide one simple evidence which I both understand and can explain clearly.

Below is what I can think of. Is this a valid argument?
Currently, we can solve GI in Quasipolynomial Time. If GI is NP-complete, then we should be able to solve other NP-complete problems in Quasipolynomial Time as well. However, right now, we are unable to solve any NP-complete problem in Quasipolynomial Time. Therefore, GI cannot be in np-complete.
The type of argument you are looking for is as follows:

If graph isomorphism were NP-complete, then some widely believed complexity assumption fails.

There are at least two such arguments:

Schöning showed that if graph isomorphism is NP-complete then the polynomial hierarchy collapses to the second level (equivalently, $\Sigma_2^P = \Pi_2^P$).

Babai's quasipolynomial time algorithm implies that if graph isomorphism is NP-complete then the exponential time hypothesis fails.


Whether such arguments are &quot;valid&quot; or not is a sociological question, not a mathematical one.