Find all edge-transitive Cayley graphs on cyclic group of valency 3.(Up to isomorphism)
My solution : So since the valency has to be equal to 3 $\Rightarrow$ the $|S|=3$. Take a cyclic group $\mathbb{Z}_n$. Then Cay($\mathbb{Z}_n$,S), S defined as before is edge- transitive on a cyclic group. And it is edge-transitive
My quation: Are graphs of such kind the only ones that are edge-transitive Cayley graphs on cyclic group of valency 3? And if yes, then how can i show it?
Since you are considering Cayley graphs, the connection set $S$ must be inverse-closed. Since it has size $3$, it must contain the unique involution $y$ of the cyclic group $G$. Let $x^{\pm 1}$ be the other two elements of $S$. We may as well assume that $G=\langle x,y\rangle$. There are two cases, depending on whether $y\in\langle x \rangle$ or not. If it is, then then graph is a Mobius ladder, if not then $x$ has odd order (otherwise $G$ would not be cyclic) and the graph is a ladder (or prism) of order twice odd. None of the graphs in the second category are edge-transitive, whereas in the first one, the first two graphs are edge-transitive (they are isomorphic to $K_4$ and $K_{3,3}$.)