  These notes concern aspects of various graphs whose vertex set is a group $G$
and whose edges reflect group structure in some way (so that they are invariant
under the action of the automorphism group of $G$). The graphs I will discuss
are the power graph, enhanced power graph, deep commuting graph, commuting
graph, and non-generating graph, though I give a briefer discussion of the
nilpotence and solvability graphs, and make some remarks on more general
graphs. Aspects to be discussed include induced subgraphs, forbidden subgraphs,
connectedness, and automorphism groups. We can also ask about the graphs formed
by the edges in one graph but not in an earlier graph in the hierarchy. I have
included some results on intersection graphs of subgroups of various types,
which are often in a dual relation to one of the other graphs considered.
Another actor is the Gruenberg--Kegel graph, or prime graph, of a group: this
very small graph influences various graphs defined on the group. I say little
about Cayley graphs, since (except in special cases) these are not invariant
under the automorphism group of $G$.
  The graphs all have the property that they contain \emph{twins}, pairs of
vertices with the same neighbours (save possibly one another). Being equal or
twins is an equivalence relation, and the automorphism group of the graph has a
normal subgroup inducing the symmetric group on each twin class. For some
purposes, we can merge twin vertices and get a smaller graph. Continuing until
no further twins occur, the result is independent of the reduction, and is the
$1$-vertex graph if and only if the original graph is a \emph{cograph}. So I
devote a section to cographs and twin reduction, and another to consequences
for automorphism groups.
  There are briefer discussions of related matters.
