  If $v$ is an eigenvector for eigenvalue $\lambda$ of a graph $X$ and $\alpha$
is an automorphism of $X$, then $\alpha(v)$ is also an eigenvector for
$\lambda$. Thus it is rather exceptional for an eigenvalue of a
vertex-transitive graph to be simple. We study cubic vertex-transitive graphs
with a non-trivial simple eigenvalue, and discover remarkable connections to
arc-transitivity, regular maps and Chebyshev polynomials.
