For any simple, finite, undirected graph $G=(V,E)$ and $v\in V$ let $N(v) = \{w\in V:\{v,w\}\in E\}$.

Suppose $G, H$ are finite, simple, undirected graphs and there is a bijection between the vertex sets $\varphi:V(G) \to V(H)$ such that for all $v,w\in V$ (not necessarily distinct) we have

$$|N(v)\cap N(w)| = |N(\varphi(v))\cap N(\varphi(w))|.$$

Does this imply that $\chi(G) = \chi(H)$?