Since this stuff keeps growing, I have decided to keep the most recent version on the web. It is now nearly twice as long as the first version I circulated. If you are interested, please take a copy: it’s at http://www-groups.mcs.st-andrews.ac.uk/~pjc/graphs/graphs.html
What is new in this version? A section about cographs and twin reduction has been extended, with the results of computation on small simple groups; there is a note about Cayley graphs invariant under the automorphism group of the group; and a few brief comments about other graphs such as the nilpotency and solvablility graphs of a group, graphs on semigroups, and graphs on rings.
Peter Cameron's Blog 
This is getting obsessive. Two days after I posted that, I managed to prove a result about the graph where g and h are joined if they commute but neither is a power of the other. If the GK graph of G is connected, and an extra hypothesis holds, then this graph (on the non-identity elements) either has an isolated vertex or is connected. The extra hypothesis is stronger than I would like: I need to assume that every Sylow subgroup of G has non-cyclic centre. Surely we can do better…
Putting the paper on the web didn’t cure my obsession, so I have put it on the arXiv. That probably means that the link given above will not be so regularly updated.
This has had one intended effect: I have had several very valuable pieces of information from various people since it went on the arXiv, which will certainly be incorporated in the next version (whenever that happens).