My first question:
Let $G$ be a graph that is both vertex-transitive (Then $G$ is regular of degree k, say.) and edge-transitive , prove that $G$ is bipartite and its simple eigenvalue is $k$ and $-k$.
Another question: if $G$ only has simple eigenvalue , prove that $G$ has at most two vertices.
I find that a triangle is vertex-transitive and edge-transitive ,but it's not bipartite. So how can I correct the first problem? For the second problem, a simple eigenvalue $\theta$ must satisfy $\theta\equiv k\; (\mod\;2)$. Since $G$ is $k-$regular, $|\theta|\leq k$. If $G$ is a simple graph ,then the problem is done, but if $G$ is not simple ,how can I solve it ? Any help appreciated.