Longest path among all pairwise shortest paths in unweighted undirected biconnected planar graphs Consider an unweighted undirected bi-connected planar graph.

Let $l_{v,u}$ be the length of the shortest path between nodes $v$ and $u$. 

Let $l_{max}$ be the length of the longest shortest path from any node to any other node.

The question is:


  Are there any known non-trivial upper/lower bounds for $l_{max}$?


It would also help if the bounds are for more general types of graphs.

Another interesting way of looking at the problem is:


$l_{max}$ is the maximal number of hops it takes to send a packet in a network from any terminal to any other, when using the shortest possible path.


Please post any link/reference that might be relevant, even if it does not directly solve the problem.
Thanks in advance!
$l_{max}$ is called diameter of a graph.
I hope this paper will be helpful