what is the maximum number of vertices in a regular graph with fixed diameter

Question

Suppose a $k$-regular graph G of diameter $d$. I am going to prove $|V(G)|=n$, $n\le [k(k-1)^d-2]/(k-2)$ if $k > 2$ I need some hint for it. Thanks

Answer 1

Hint: Fix any node as a root, branch $k$ ways, and repeat for each of the children, branching $k-1$ ways for them. After the root node at level $r=0$, there are $k(k-1)^{r-1}$ nodes for level $r>0$, up to level $r=d$. Now sum a finite geometric series.