Let $G$ be a connected graph on $n$ nodes, $m$ edges. A graph is $(b, r)$ coverable if it can be covered by $b$ balls of radius $r$ (in other words, there is a set of $b$ nodes such that all nodes are within distance $r$ of one of these $b$ nodes). Suppose $G$ is $(b, r)$ coverable.
Intuitively, I feel like there should be a nice relationship between $m, b,$ and $r$. For example, if $m$ is set to its minimum value ($n-1$), then we have the relationship $b(2r+1) \ge n$ (because the nodes are arranged in one big chain). Alternately, if $m$ is set to its maximum value ($n(n-1)$), then $G$ is the complete graph and it is trivially $(1, 1)$ coverable.
Is there some bound of the form $f(b, r, m) \ge g(n, m)$ that generalizes these examples?
If your graph is $k$-regular, then you can approximate diameter with $log_{k}(n)$. Let $\delta(G) := min \{ d(v) : v \in V(G) \}$. You can bound the diameter above by $\lceil log_{\delta(G)}(n) \rceil$. Similarly, let $\Delta(G)$ be the maximal degree of the graph, and you can use it to get a lower bound on the diameter. Make sure to floor the lower bound.