Graphs defined on groups

Abstract

‎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.