Are there $2^{\aleph\_0}$ pairwise non-isomorphic connected [vertex-transitive](https://en.wikipedia.org/wiki/Vertex-transitive_graph) graphs $G$ with $V(G) = \omega$?



Non-isomorphic connected vertex-transitive graphs on $\omega$
Yes. In particular, there are continuously many pairwise non-quasi-isometric Cayley graphs with countably infinite vertex set. A proof of this is in the paper *Continuously many quasi-isometry classes of 2-generator groups* by Bowditch.


