# Are there any connected graphs with constant link which are not vertex transitive?

By constant link, I mean for any vertices $v,w$ of a graph $G$, the subgraph of $G$ induced by the neighborhood of $v$ is isomorphic to the subgraph induced by the neighborhood of $w$.

$C_n + C_m$ satisfies these conditions for $m \neq n$, which I was the first example that I was able to come up with, but I'm finding it difficult to conceive of a connected graph with constant link which is not vertex transitive.

Is anything known on the existence or lack thereof of such graphs?