Thank you! It’s actually exciting to see that someone found this well over two years later.

In spite of the paper’s title, it’s really a paper in category theory and not algebra or graph theory. Basically I came up with a theorem in category theory that I’ve never seen before, but it requires some specific adjunctions to apply it. I did a little digging to see if I could find any interesting cases where it worked, and by far the most interesting one I came up with used the adjunction between the commutation graph functor and the right-angled Artin group functor. I felt that the application was more noteworthy than the big theorem where the heavy lifting was done, so I used that as the title. Other than understanding what a free group is, there’s virtually nothing on graphs or groups in the paper other than their categories having the specific properties I needed, which I just cited from other references.

The introduction discusses some stuff about frames and locales which is only related in that it gives context for the genesis of the big theorem. Basically, you won’t be able to read the introduction if you’re not comfortable with locales, lattices, and topological spaces, but if you know some basic category theory you can read the rest of the paper without the introduction.

I don’t know for sure my theorem is new, it’s definitely possible that it’s an immediate corollary of some other existing theorem that I’m just totally unfamiliar with. It’s unfortunately in that sweet spot where it’s way too advanced to be read by anyone not deep in math, but way too elementary for someone serious about math to care about it. Somehow I’m still proud of it though, I think it’s a lot better than any of the research I did when I was still in school.