Expected Diameter of an Undirected Graph Given the Number of Nodes and Edges

Question

This question is similar to this question I asked earlier. In this question, however, I would like to know if there is a way to determine the expected diameter of a undirected unweighted graph given the number of nodes, $V$, and the number of edges, $E$?

Answer 1

Although the two questions sounds similar, it is completely different because we have a pretty good idea of what is a typical random graph -whereas in the other question, you must consider all graphs, the 'interesting' graphs will be dwarfed in number by the typical ones. The uniform random graph conditioned on having $n$ vertices and $m$ edges is called the Erdős–Rényi model, and often written $G(n,m)$. It is closely related to its binomial counterpart $G(n,p)$, where each edge is present with probability $p(n)$, and results can be transferred from one model to the other with some care.

Now back to the expected diameter, obviously the results differ wildly depending on the value of $m(n)$/$p(n)$, but there are pretty good bounds holding with high probability that are known. You should know where to look for with that information.