Fix $n$, and consider the characteristic polynomials for all $C=2^{\frac{n(n-1)}{2}}$ adjacency matrices representing undirected, unweighted graphs on $n$ vertices.
Are the characteristic polynomials distributed somewhat uniformly among adjacency matrices? Or are some characteristic polynomials "much more popular" than others? If so, is there some Zipf-law distribution?
The work of Andries Brouwer counts the number of characteristic polynomials for smallish $n$ relative to the number of graphs.
Similarly fix $n$, and consider the Alexander polynomials associated with the $C=n!\times \lfloor{\frac{n!}{e}}\rfloor$ grid diagrams that represent various links/knots.
Are the Alexander polynomials distributed somewhat uniformly among the grid diagrams?
The work of Canterralla, Chapman, and Mastin suggests that a Zipf-like law tends to apply for random knot diagrams - some knot types are more popular than others. I suspect that there are more grid diagrams that have an Alexander polynomial associated with, say, $7_6$ than with $7_1$, because the symmetries of $7_1$ make it harder to be randomly generated. But if $n$ were large enough, maybe uniformity takes over.
My motivation considers a uniform superposition $\frac{1}{\sqrt{C}}|A \rangle|B \rangle$ - $A$ being over adjacency matrices (grid diagrams) and $B$ being a characteristic polynomial (Alexander polynomial) of the respective adjacency matrix (grid diagram). Over all $|A \rangle$, is $|B \rangle$ most likely uniformly distributed? If I measure $|B\rangle$ will some polynomials be more likely than others?
