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Are there $2^{\aleph_0}$ pairwise non-isomorphic connected vertex-transitive graphs $G$ with $V(G) = \omega$?

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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.

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    $\begingroup$ I think this result is originally due to Grigorchuk (maybe be with 4 rather than 2 generators). $\endgroup$
    – YCor
    Commented Mar 8, 2022 at 14:03
  • $\begingroup$ This follows also e.g. from the existence of Cayley graphs with a wide range of growth behaviors, which is a quasi-isometric invariant. $\endgroup$
    – M. Winter
    Commented Mar 8, 2022 at 16:48
  • $\begingroup$ @M.Winter yes this is precisely Grigorchuk's argument. $\endgroup$
    – YCor
    Commented Mar 8, 2022 at 20:49

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