Graph auto-layout algorithm

I would say as Noufal Ibrahim, but you could also look more precisely at the C API of the graphviz project. It includes a lib to build your graph (libgraph.pdf) with all the nodes and edges, and a lib to layout the graph (libgvc.pdf) (just compute each nodes position), so you can then display it in your own UI for example.

I would suggest that you take a look at at graphviz. The dot program can take a specification of a graph and generate an image of the network for you somewhat "cleanly". I've linked to the "theory" page that gives you some links that might be relevant if you're interested in the theoretical background. The library and tools themselves are mature enough if you simply want a solution to a problem with layout that you're facing.

You will find http://graphdrawing.org/ and this tutorial, by Roberto Tamassia, professor at Brown University, quite helpful.

I like a lot Force-Directed Techniques (pp. 66-72 in the tutorial) like the Spring Embedder.

You assume there is a spring or other force between any two adjacent nodes and let nature (simulation) do the work :)