Some work in this direction appears in [@best08].

On the other hand, some graphs are known not to admit uniform mixing:

(a) $K_n$ for $n > 4$.

(b) $H(n,q)$ for $q > 4$.

(c) Cycles $C_n$ for $n = 2^uq$, where $q = 1$ ($u \geq 3$) or $q \cong 3 \mod 4$ [@adamczak07].

(d) The cycle $C_5$ [@carlson07].

(e) Complete multipartite graphs (except $C_4$) [@ahmadi03].

(f) The transpositions Cayley graph $X(S_n,\{\text{transpositions}\})$ [@gerhardt03].

An obvious question raised by these results is which cycles admit uniform mixing. In this work we show that the cycle $C_n$ does not admit uniform mixing if $n$ is even and greater than four, or if $n$ is a prime greater than three. We see that if a bipartite graph on $n$ vertices admits uniform mixing then $n$ must be divisible by four; if in addition the graph is regular then $n$ must be the sum of two integer squares.

The graphs we have that do admit uniform mixing are highly regular, and this leads us to ask which strongly regular graphs admit uniform mixing. We use work of Chan [@chan08] to show that the only strongly regular graphs that admit instantaneous uniform mixing are the 