User blog comment:QuasarBooster/Fibonacci/Lucas Sequence Extentions?/@comment-2033667-20150730151541/@comment-1605058-20150730215407

Well, $$\phi$$ is a root of $$x^2-x-1$$ of maximal absolute value, which is why it is arising in Fibonacci numbers.

If you are asking if $$\phi$$ is directly related to maximal roots of other polynomials, then I'm afraid the answer is no, though I have no way to argue about that.