Q:

In a graph $G$, $d(u,v)=m$.Find $d(u,v)$ in $G^n$

Let $G$ be a graph . we define $G^{n}$ as: $V(G)=V(G^{n})$ and for $v,u \in V(G)=V(G^{n}) $ ,$u$ and $v$ adjacent in $G^{n}$ if $d(u,v)\le n$.Now suppose in some graph $G$ , $d(u,v)=m$.Find $d(u,v)$ in $G^{n}$.

A:

Use long division to write $m=qn+r$, $0\leq r\leq n-1$.
Let $(v_0=u,v_1,v_2,\ldots,v_{m-1},v_m=v)$ be a shortest path of length $m$ from $u$ to $v$.
If $r=0$ then $v_0,v_n,\ldots,v_{qn}=v_m$ is a path of length $q$ in $G^n$.
If $r>0$ then $v_0,v_n,\ldots,v_{qn},v_m$ is a path of length $q+1$ in $G^n$.
This argument is valid for both $m>n$ and $m\leq n$ if you are willing to accept the "degeneration" of the mentioned sequences.
Strictly spoken you also need an argument that a shorter path is impossible, but that is rather boring.
(Would have done this as a remark if I had the reputation, since it does not add a lot, but everything until now was at least incomplete.)

