Consider a Cayley graph $G=\mathbb Z/n$, under addition in $\mathbb Z$. Let $M$ be the set of generators such that $a\in M \implies -a \in M$. Suppose there are $m$ generators, so we may assume $G$ is an m-regular graph with $n$ vertices. For any vertex $v$, I 'm interested in finding an upper bound of the number of vertices that are $l$ distance away from $v$.
For example, for $l=1$, there are exactly $m$ vertices. For $l=2$, there will be fewer than $m(m-1)$ vertices that are 2-distance away from $v$. Follow this, we may conclude the number of vertices that are $l$ distance away from $v$ is smaller than $m(m-1)^{l-1}$. However, this estimate may be too loose. Is there a better upper bound I can use in this case?