Let $G(V,E)$ be a graph , suppose that $\vert N_{i,r} \vert $ be number of vertices whose geodesic distance from vertex $i$ is exactly $r$ edges.if $d_i$ the degree of vertex $i$ and $d_{max} := \max_n {d_n}$, how can resulted that
$$\vert N_{i,r} \vert \leq d_i (d_{\max} -1)^{r-1} ?$$
HINT:
The short answer is that this follows by induction on $r$; it's clearly correct for $r = 1$, because the number of vertices whose distance from $i$ is no more than $1$ is certainly bounded by the degree, $d_i$, of vertex $i$. (And if you only allow one edge between any two vertices, then it's not only bounded but equal to $d_i$).
Now you just need to work out the inductive step. When you do so, you can show your progress by clicking on the "edit" link below your question. And if you get stuck, we can help you from there.