how to construct graph $G$ where $\operatorname{Aut}(G)\simeq \mathbb{Z}_n$

Question

for group $\mathbb{Z}_n$ (arbitrary $n$) I want to make a graph $G$ where $\operatorname{Aut}(G)\simeq \mathbb{Z}_n$, for $n=2$ it is $P_n$ and $K_2$ (if I am not wrong), how should I construct $G$?

Answer 1

The pdf Uma linked to in the comments gives the idea, but I'll lay it out here. Basically we want an $n$-cycle, but we don't want there to be reflection symmetry, so take a directed $n$-cycle and replace every directed edge $$\bullet \to \bullet$$ with the undirected graph $$\bullet - \overset{\overset{\overset{\overset{\bullet}{|}}{\bullet}}{|}}{\bullet} - \overset{\overset{\bullet}{|}}{\bullet} - \bullet$$ It's not hard to see that a graph automorphism must preserve these pieces and their orientation, so this doesn't change the automorphism group.