Let's call a simple, undirected graph $G=(V,E)$ *homogeneous* if for every $v,w\in V$ there is a graph isomorphism $\varphi:G\to G$ such that $\varphi(v)=w$.


It is clear that every finite homogeneous graph is $k$-regular for some $k\in\mathbb{N}$.


Is there $k\in\mathbb{N}$ and a connected $k$-regular graph that is not homogeneous?


(The smallest example of a $2$-regular graph that is not homogenous is the disjoint union of $K\_3$ and $C\_5$, but in this question I'm focusing on connected graphs.)



Connected homogeneous graphs
Homogeneous is usually called *vertex transitive.* Not all connected regular graphs are vertex transitive. For example, the [Frucht graph](https://www.wikiwand.com/en/Frucht_graph) is a 3-regular connected graph that is not vertex transitive (its automorphism group is the identity). 


