This paper is complete nonsense. A massive claim is made, but the presented time complexity analysis consists of "...seems to fall roughly into Quadratic Time with...".

Note that I (and anyone else) can routinely solve NP-complete problem instances in quadtratic time. Travelling salesman and SAT instances instances with n = 100,000 are often tractable with modern algorithms. What I can't do is guarantee that I can solve EVERY instance in quadratic time. This paper gives a simple method that works well for the 100 instances the author tried. It does not show that all CLIQUE instances can be solved in polynomial time.

The logic seems to be: CLIQUE is NP-complete as a class of problems, so it takes 2^n steps to solve any CLIQUE instance. Hence if I find a method that does better on the first few instances I look at, I have shown that my method is polynomial for the all CLIQUE instances. This is garbage.

I also note from the text "...This paper did not provide a Polynomial Time Solution to the Clique problem as the author claimed. The author was in error (I know because I am him) and made the faulty assumption...". So the author has a track record of deluding himself and disseminating the resulting nonsense.

Thanks for your reply. It claims n^2 time to find the optimal answer to any clique instance (not 2^n). The paper optimally prunes off non-symmetries leaving a fundamental core. Please send along an instance that you feel it will not optimally solve. -A.

That's not how it works. You are making the claim for all instances. So you should be able to reproduce something like Section 4 from (for example)

https://www.baeldung.com/cs/matrix-multiplication-algorithms

This shows how multiplying ANY two matrices is polynomial time. No point looking for an instance where it fails, as it covers all possible cases.

You shouls also look at this:

Article Why Is Maximum Clique Often Easy in Practice?

What you have done simply confirms the findings of these excellent (and peer-reviewed) findings.

Thank-you for the links. I am familiar with the 2nd, my methods are very different. Yes, I am making the claim for all instances ...and yes, I clearly need to enhance the time complexity analysis. Nobody will peer review my (a non-academics) work...especially with this claim. Why do you think I am here. Any submissions with an N beside a P to a Math or Comp.Sci. Journal from a non-institution affiliated individual are programmed to go directly into their spam folder. Time will tell. Best -A.