Document Type : Research Paper
School of Mathematics and Statistics, University of St Andrews, UK
This paper concerns aspects of various graphs whose vertex set is a group $G$ and whose edges reflect group structure in some way (so that, in particular, they are invariant under the action of the automorphism group of $G$). The particular graphs I will chiefly discuss are the power graph, enhanced power graph, deep commuting graph, commuting graph, and non-generating graph.
My main concern is not with properties of these graphs individually, but rather with comparisons between them. The graphs mentioned, together with the null and complete graphs, form a hierarchy (as long as $G$ is non-abelian), in the sense that the edge set of any one is contained in that of the next; interesting questions involve when two graphs in the hierarchy are equal, or what properties the difference between them has. I also consider various properties such as universality and forbidden subgraphs, comparing how these properties play out in the different graphs.
I have also 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 has a surprising influence over various graphs defined on the group.
Other graphs which have been proposed, such as the nilpotence, solvability, and Engel graphs, will be touched on rather more briefly. My emphasis is on finite groups but there is a short section on results for infinite groups. There are briefer discussions of general Aut(G)-invariant graphs, and structures other than groups (such as semigroups and rings).
Proofs, or proof sketches, of known results have been included where possible. Also, many open questions are stated, in the hope of stimulating further investigation.