Simple eigenvalues of cubic vertex-transitive graphs

If v is an element of R(V(X))is an eigenvector for eigen value 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 eigen value of a vertex-transitive graph to have multiplicity one. We study cubic vertex-transitive graphs with anontrivial simple eigenvalue, and discover remarkable connections to arc-transitivity, regular maps,and number theory