Just in case anyone else is still thinking about this question...

The answer is the following. Either:

 1. The eigenvalues of $G_n$ are all of the form $\zeta + \zeta^{-1}$ for roots of unity $\zeta$, and the graphs $G_n$ are subgraphs of the Dynkin diagrams
$A_n$ or $D_n$.

 2. For sufficiently large $n$, the largest eigenvalue $\lambda$ is greater than two, and
$\mathbf{Q}(\lambda^2)$ is not abelian.

The proof is effective, but a little long to post here. The main ingredients are some basic facts about Weil height, some ideas due to Cassels, and an amplification step using Chebyshev polynomials.

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The [paper][1] is now on the ArXiv.  See the last section for the proof of this result, and the second to last section for a more effective but logically weaker result.


  [1]: http://arxiv.org/abs/1004.0665