I have a graph $G=(V,E)$, where $V$ is a vertex set, and $E$ is an arc set. Now given a vertex $i\in V$, I want to find the (small) upper bound for the number of the nodes in $V$ that has the length of $l$ from $i$.
For example, when $l=1$, we can find the degree of vertex $i$, $deg(i)$, is the good upper bound for the number of the nodes in $V$ that has the length of $1$ from $i$.
When $l=2$, maybe $deg(i)*deg_{\max}$ is the upper bound for the number of the nodes in $V$ that has the length of $2$ from $i$, but $deg(i)*deg_{\max}$ seems a bit larger bound.
For the general case $l$, are there any small upper bounds ? Thanks.
An upper bound is $\deg(i) (\deg_\max - 1)^{\ell-1}$. This is best possible in the sense that it is exact in the case of a tree where $i$ is the root, all leaves are at distance $\ell$ from the root, and all nodes other than the root and the leaves have degree $\deg_\max$.