In Mathematica 10 we have the new FindCycle function. Maybe something like this could help:

g=Graph[myList];
sg=Subgraph[g,#]&/@WeaklyConnectedComponents[g];
DeleteCases[FindCycle[#,?,?]&/@sg,{}]

Hi @Szabolcs, for:

g = Graph[{1 -> 2, 2 -> 3, 3 -> 2, 2 -> 1}]

{{{2-> 3, 3 -> 2}, {1-> 2, 2->1}}}

Hi @Sander, you can easily count it, and retrieve it using MaximalBy. FindCycle will show all of then.

In the old Combinatorica package, documented in "Computational Discrete Mathematics: Combinatorics and Graph Theory with Mathematica" by Pemmaraju and Skiena, on page 332 they construct the Eccentricity of a vertex as the length of the longest shortest path from a vertex v to any other vertex. They then construct the Diameter as the Max of Eccentricity over all verticies v. Perhaps you can study these and adapt them to your task.

<< Combinatorica`

and then

?? Diameter

and

?? Eccentricity

I don't know how well this is going to apply to a problem with tens of thousands of verticies and edges.

If you are going to do anything substantial with graphs then I'd start with the book. Surprisingly, there are still used copies available that the book vultures have not snapped up and now want a thousand dollars for.