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.