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    Simple eigenvalues of cubic vertex-transitive graphs

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    If vv is an eigenvector for eigenvalue λ\lambda of a graph XX and α\alpha is an automorphism of XX, then α(v)\alpha(v) is also an eigenvector for λ\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.Comment: 22 p
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