Let $G$ be an edge-transitive and $k$-regular graph and $\lambda$ be its simple eigenvalue.
Show that $\lambda = \pm k$.
Definitions:
$G$ is edge-transitive if for any two edges $e$ and $e'$, there exists an antomorphism of $G$ mapping $e$ to $e'$.
Simple eigenvalue of $G$ is an eigenvalue of the adjacency matrix $A(G)$ of $G$ with multiplicity 1.