Why does no undirected graph has eigenvalue $\sqrt{2+\sqrt{5}}$?

Question

Here is the hint: Consider the algebraic conjugates of this number. But I don't know how to use it. Any help is appreciated!

Answer 1

The adjacency matrix is a symmetric real matrix, hence all eigenvalues are real (spectral theorem). But the matrix is also rational, hence any algebraic conjugate of an eigenvalue is again an eigenvalue. Among the algebraic conjugates of $\sqrt{2+\sqrt 5}$ is $\sqrt{2-\sqrt 5}=\sqrt{\sqrt5-2}\,i\notin\Bbb R$.