  We prove an inequality on the difference between the spectral radius of the
Cayley graph of a group $G$ and the spectral radius of the Schreier graph
$H\backslash G$ for any subgroup $H$. As an application we extend Kesten's
theorem on spectral radii to uniformly recurrent subgroups and give a short
proof that the result of Lyons and Peres on cycle density in Ramanujan graphs
holds on average. More precisely, we show that if $\mathcal G$ is an infinite
deterministic Ramanujan graph, then the time spent in short cycles by a random
walk of length $n$ is $o(n)$.
