Is there anything special about a graph with the golden ratio in its spectrum?

Question

Given a simple connected graph $g$ with adjacency matrix $\mathbf{A}$. Let the spectrum $\lambda_1 < \lambda_2 < \ldots < \lambda_N$ be the eigenvalues of the equation $\mathbf{A} v=\lambda v$. For lack of a better word, let a graph be called golden if there is an eigenvalue such that $\lambda_i = n \phi$ where $n \in \mathbb{N}, |n| > 0$ and $\phi$ is the golden ratio.

Do golden graphs have any special symmetry or property? I computed all simple connected golden graphs of order 6 as a starting point:

Answer 1

The experimental evidence indicates that "most" graphs have characteristic polynomial irreducible over the rationals. Thus most graphs do not have the golden ratio as a root. Note also that we do not have a nice characterization of graphs with 1 (say) as an eigenvalue, perhaps we should not expect too much for the golden ratio.

Answer 2

The minimal polynomial of $n \phi$ is $z^2 - n z - n^2$, so a graph is "golden" iff its characteristic polynomial is divisible by $z^2 - n z - n^2$ for some nonzero integer $n$.