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

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, $E$?

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.
• To the interested reader: Here is a paper by Fan Chung and Linyuan Lu which surveys results on likely diameter for $G(n,p)$ for various ranges of $p$. It is also has a nice table of how concentrated the likely diameter is. Some of the results for $np=c$ for $c$ close to 1 have been improved since then. – D Poole Sep 12 '16 at 14:23