Given a connected graph $G =(V,E)$ and a positive integer $k$, the k-th power of $G$, denoted $G^k$ , is the graph whose set of nodes is $V$ and where vertices $u$ and $v$ are adjacent in $G_k$ if and only if $d(u,v) \le k$ in $G$.
$P_8$ is a path graph with 8 vertices such as: o-o-o-o-o-o-o-o
$C_{10}$ is a cycle graph: $C_{10}$ has 10 vertices.
$d(u,v)$ is the distance between $u$ and $v$, which is the length of the shortest path $u-v$ from $u$ to $v$ in graph $G$.
I know that two vertices are adjacent in the 2nd power if and only if there's a path of length at most 2 between them in $P_8$.
- Draw the 2-nd and 3-rd powers of $P_8$ and $C_{10}$ .
- For a graph $G$ of order $n$, what is $G^{diam(G)}$
I know the stuff above and I am asking for any answers and help, appreciate it.