no cycle $C_n$, except for $C_3$ and $C_4$, admits uniform mixing. Adamczak et al. [@adamczak07] show that $C_n$ does not admit uniform mixing if $n = 2^u$ for $u \geq 3$ or if $n = 2^{u}m$ where $m$ is not the sum of two integer squares and $u \geq 1$. Carlson et al. [@carlson07] show that uniform mixing does not occur on $C_5$. (This result also follows from the earlier result in this paper that uniform mixing does not occur on conference graphs.)

Using Theorem \[no\_algebraic\], we show that uniform mixing cannot occur on any even cycle. Likewise, we see that uniform mixing cannot occur on $C_p$ where $p$ is a prime such that $p \geq 5$. For both of these results, we use the irrationality of the eigenvalues of the underlying graph.

Recall that the eigenvalues of a general cycle $C_n$ have the form $$\theta_r = \zeta^r + \zeta^{-r},$$ where $\zeta = e^{2 \pi i /n}$. If $n$ is even, then $\theta_0 = 2$ and $\theta_{n/2} = -2$ are both simple eigenvalues of $C_n$, and the remaining eigenvalues each have multiplicity 2. On the other hand, if $n$ is odd, then $\theta_0 = 2$ is the only simple eigenvalue, 