Digraphs on groups

I have spent a lot of time recently thinking about graphs on groups. To recall the rules: the vertex set must be the group (in general, but not here, I allow an automorphism-invariant subset or the quotient by an automorphism-invariant equivalence relation); the graph must be defined in terms of the group structure with no arbitrary choices; so the automorphism group of the group acts as automorphisms of the graph. There are many many examples of graphs on groups. (Some examples: power graph,...