Cayley Graph and cyclic groups

Question

If $G$ is a finite cyclic group then Cayley $G((a, a^{-1}))$ is a cycle of order $n$. I have done this but I am unable to crack the converse of it. Please help.

Answer 1

Let $H$ be a directed cycle graph (so that no two vertices are in opposing direction) on $n$ vertices. Let $G$ be a cyclic group of order $n.$ Let $\langle x \rangle = G,$ be any generator. Label the vertices of $H$ with the elements of $G,$ and then label the edges with $x,$ and you've got your cycle graph correspondence.