and the nontrivial eigenvalues each have multiplicity two.

\[irrational\_lemma\] If $n =5$ or $n \geq 7$, then $C_n$ has an irrational eigenvalue.

[*Proof.* ]{}Using the notation above, we note that $\theta_1 = 2 \cos(2 \pi i / n)$. Since $n$ is a positive integer, the eigenvalue $\theta_1 = 2 \cos(2 \pi i / n)$ is rational if and only if $n \in \{1,3,4,6\}$ (see [@olmsted45].) If we assume $n = 5$ and $n \geq 7$, then $\theta_1$ is an irrational eigenvalue of $C_n$.[  ]{}

Using our work on bipartite graphs and Theorem \[no\_algebraic\], we eliminate the possibility of uniform mixing occurring on all even cycles other than $C_4$. This extends the work of Adamczak et al. [@adamczak07].

The cycle $C_4$ is the unique even cycle that admits uniform mixing.

[*Proof.* ]{}Let $U(t)$ denote the transition matrix of $C_{2m}$. If $m = 2$, it is straightforward to observe that $U(t)$ is flat at time $t = \pi/4 + r \pi/2$ for any $r \in {{\mathbb Z}}$. Suppose for a contradiction that $m > 2$ and $U(t)$ is flat at some time $t$. From the discussion in Section \[bipartite\_graphs\], it follows that $m$ must be even, and so $m > 3$. Furthermore, 