Let $G$ be a connected (strongly connected) graph (digraph).
Assume that the minimal vertex degree (in/out degrees) of the graph is $\delta$ (are $\delta^-,\delta^+$).
What is the maximal diameter possible for such graph?
For example, if $\delta \geq \frac {n}{2}$ ($\delta^- + \delta^+ \geq n-1)$, then the graph is of diameter at most 2.
Also, it seems that $\delta \geq c\cdot n$ gives an upper bound of $\lceil \frac{2}{c} \rceil - 1$ on the diameter, but I'm not sure it's tight (also, haven't tried proving it yet, so I might be wrong).
(For $c=\frac{1}{2}$ it's definitely not tight).
What other bounds can we get?