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If 
${\mathbf v} \in {\mathbb R}^{V(X)}$ is an eigenvector for eigenvalue 
$\lambda $ of a graph X and 
$\alpha $ is an automorphism of X, then 
$\alpha ({\mathbf v})$ is also an eigenvector for 
$\lambda $. Thus, it is rather exceptional for an eigenvalue of a vertex-transitive graph to have multiplicity one. We study cubic vertex-transitive graphs with a nontrivial simple eigenvalue, and discover remarkable connections to arc-transitivity, regular maps, and number theory.
