Many connected vertex-transitive graphs $G=(V,E)$ have the property that some of their automorphisms other than the identity have fixed points. To point out two simple examples:
- If $G = K_3$ then the automorphism swapping the points of an edge and leaving the remaining point intact has $1$ fixed point.
- For $G = C_4$, the "mirror map" along one of the diagonals has those diagonal points as fixed points.
But for both graphs, there are automorphisms that do not have any fixed points. (For both examples, consider a rotation map.)
Is there a connected vertex-transitive graph $G=(V,E)$ with $|V| \geq 2$ such that every automorphism has a fixed point?