# Imposing reciprocity in the definition of vertex-transitivity

A simple, undirected graph is vertex-transitive if for any pair of vertices $$x,y$$, there exists an automorphism (adjacency-preserving self-bijection) $$\phi$$ such that $$\phi(x)=y$$.

What if, instead of taking $$x$$ to $$y$$ as above, we require the automorphism $$\phi$$ to exchange $$x$$ and $$y$$, i.e. $$\phi(x)=y$$ and $$\phi(y)=x$$?

1. Is there a name for this natural refinement of the notion of vertex-transitivity?
2. What is a simple example of a vertex-transitive graph which does not satisfy this?

Note that any Cayley graph whose generating set is conjugacy-invariant does satisfy this exchange property (take $$\phi(u)=xu^{-1}y$$).

• Generously transitive is the name for this property. Feb 27 at 10:32
• See math.stackexchange.com/questions/953856/… for discussion of this property. Feb 27 at 10:33
• Thank you so much, Gordon! The name "generously transitive" is not very natural, so I would have had a hard time finding it by myself. I am new to this site, what shall I do to validate/close?
– DRJ
Feb 27 at 12:58
• @GordonRoyle, since @‍DRJ seems satisfied with your comment, would you post it as an answer so that they can accept it? Feb 27 at 21:56
• @DRJ To close the question, you accept the answer that I have now given as an actual answer, rather than a comment. Feb 28 at 1:36

A permutation group $$G$$ acting on a set $$X$$ is called generously transitive if for any two elements $$x$$, $$y \in X$$ there is a permutation $$g \in G$$ such that $$x^g = y$$ and $$y^g = x$$.