Suppose that there exist two connected graphs $G$ and $H$ and a one-to-one function $\varphi$ from the vertex set $V(G)$ onto $V(H)$ such that the distance $\operatorname d_G(u, v) = \operatorname d_H(\varphi(u), \varphi(v))$ for every two vertices $u$ and $v$ of $G$. Prove or disprove: $G$ and $H$ are isomorphic.
Hey guys, I really need some help. I am not sure if the distances in $G$ and $H$ being the same would justify something is isomorphism. I feel it wouldn't tell us if we possibly have cycles, are bipartite etc. I'm having a lot of trouble proving that it is false, if the statement is false.