graph edge generators

[William Clifford]

I've become interested in basic graphs and networks and I'm wondering
about what algorithms are there for generating basic regular graphs
like the simplex graph or dodecahedron graph, etc (I'm sure there are
many). I'm particularly keen on understanding the very basic functions
for determining edges in the graphs. If one didn't want the complete
graph but just a function generates the edges connected to a given
node.

I've been surfing around for this sort of info, but I'm having trouble
finding stuff at my level. If anyone knows of any resources or
tutorials or that sort of thing, I'd like to hear about those too.

Thanks!

--
William Clifford