A graph is vertex transitive if $x \mapsto y$ by an automorphism. Let $P$ denote the stronger property that $x \mapsto y \mapsto x$ by an automorphism.

Simple facts:

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

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

• $P \not \rightarrow$ Cayley. Petersen graph is an example.

Questions:

1. In addition to Petersen graph which is finite, is there an infinite connected graph with finite degree satisfying $P$ and which is not a Caley graph?

2. Is there a common name for this propery $P$? I looked a bit and didn't find it.

Thanks!