First we recall a useful formulation of the Gelfond-Schneider Theorem. This formulation is due to Michel Waldschmidt [@burger04].

\[gelfond\_schneider\] If $x$ and $y$ are two nonzero complex numbers with $x$ irrational, then at least one of the numbers $x$, $e^{y}$, or $e^{xy}$ is transcendental.[  ]{}

The above theorem is crucial in the proof of the following result.

\[no\_algebraic\] Let $X$ denote a graph. If at time $t$ all entries of the transition matrix $U(t)$ are algebraic, then the ratio of any two eigenvalues of $X$ must be rational.

Let $A$ denote the adjacency matrix of $X$, and let $\{\theta_0, \dots, \theta_d\}$ denote the eigenvalues of $A$. We consider a fixed pair of eigenvalues $\theta_r$ and $\theta_s$. Since $A$ is an integer matrix, the characteristic polynomial of $A$ is a monic polynomial with integer coefficients. Therefore each eigenvalue of $A$ is an algebraic integer. Furthermore, since $A$ is symmetric, we decompose $A$ as $$A = \sum_{k = 0}^{d} \theta_k E_k,$$ where each $E_k$ is orthogonal projection onto the $k$-th eigenspace. This directly implies that $$\label{scaled_transition}
    U(t) = \sum_{k=0}^{d} e^{\theta_k it} E_k.$$ From this we see that the eigenvalues of $U(t)$ are $\{e^{it \theta_0}, \dots, e^{it
\theta_d}\}$. If all entries of $U(t)$ 