If $G=(V,E)$ is a simple, undirected graph and $v\in V$, we set $N(v) = \{w\in V:\{v,w\}\in E\}$. 


Is there an integer $k>1$ and a connected $k$-[regular](https://en.wikipedia.org/wiki/Regular_graph) graph $G=(V,E)$ such that there are $v\neq w \in V$ with $N(v) = N(w)$? 


(Note that the definition of $N(\cdot)$ implies that $v,w$ cannot be adjacent.)



Regular graph such that $2$ distinct vertices have same neighborhood set
An example with $k = 4$ is the octahedron. 


