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 an finite example.

Question:
**Is there an infinite genersously 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 propery?", answered by Chris Godsil in the comments.)

Thanks!