A graph is vertex transitive if $x \mapsto y$ by an automorphism.

A graph is generously vertex transitive if $x \mapsto y \mapsto x$ by an automorphism.

Simple facts:

• GVT $\rightarrow$ unimodular. Just plug in the definition of unimodular to see why.

• Cayley $\not \rightarrow$ GVT. Free product $\mathbb Z_2 * \mathbb Z_3$ is an example.

• GVT $\not \rightarrow$ Cayley. Petersen graph is a finite example.

Question: Is there an infinite generously vertex transitive connected graph with finite degree which is not a Caley graph?

(Before edition, it was also asked: "Is there a common name for this property?", answered by Chris Godsil in the comments.)

Thanks!