Every graph with four or less vertices is planar. Every polynomial with degree four or less is solvable by radicals.
There are nonplanar graphs with five and more vertices. There are polynomials with degree five and more which are not solvable by radicals.
Could there be a way to assign a group/polynomial to any graph such that the graph is planar if and only if the group/polynomial is solvable (by radicals) or nilpotent or some other property?
I don't think the polynomial $$P_G(x)=\prod_{v\text{ vertex}}(x-\text{ number of vertices adjacent to }v)$$ works but who knows, maybe somehing similar or making permutations of vertices with the alternatig groups in some way ($A_n$ not solvable for $n\ge 5$)
If this coincidence is only a coincidence, can someone provide an argument as to why we don't expect any connection of this nature?
Thanks!