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 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.)