# Connected vertex-transitive graph with the fixed-point property

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?

• By the way, not directly related to your question, but the fixed-point property for graphs has apparently been studied by Bernd Schröder in "The Fixed Vertex Property for Graphs" (doi.org/10.1007/s11083-014-9337-5) Commented Apr 18, 2022 at 20:29

There is no such [EDIT:] finite graph $$G$$. Indeed, something stronger can be said. Suppose that a group $$\Gamma$$ acts transitively by permutations on a finite set $$X$$ , with $$\#X \geq 2$$. Then there is some $$\gamma \in \Gamma$$ for which $$\gamma\colon X \to X$$ has no fixed points. The proof is a simple application of Burnside's Lemma, see e.g. https://math.stackexchange.com/questions/106158/every-transitive-permutation-group-has-a-fixed-point-free-element.