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.

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.

The paper 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.

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 (extended) Dynkin diagrams $\widetilde{A}_n$ or $\widetilde{D}_n$.

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.

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.