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Simple eigenvalues of cubic vertex-transitive graphs
- Part of
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- Journal:
- Canadian Journal of Mathematics / Volume 76 / Issue 5 / October 2024
- Published online by Cambridge University Press:
- 10 August 2023, pp. 1496-1519
- Print publication:
- October 2024
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- Article
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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.
