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dc.contributor.authorCameron, Peter J.
dc.identifier.citationCameron , P J 2022 , ' Graphs defined on groups ' , International Journal of Group Theory , vol. 11 , no. 2 .
dc.identifier.otherPURE: 273687375
dc.identifier.otherPURE UUID: 81fe24b2-92cc-47c5-9fab-b41fe75367a9
dc.identifier.otherORCID: /0000-0003-3130-9505/work/93161518
dc.description.abstractThis 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.
dc.relation.ispartofInternational Journal of Group Theoryen
dc.rightsCopyright © 2021 University of Isfahan. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (, which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.en
dc.subjectPower graphen
dc.subjectCommuting graphen
dc.subjectGenerating graphen
dc.subjectQA Mathematicsen
dc.titleGraphs defined on groupsen
dc.typeJournal articleen
dc.description.versionPublisher PDFen
dc.contributor.institutionUniversity of St Andrews.Pure Mathematicsen
dc.contributor.institutionUniversity of St Andrews.Centre for Interdisciplinary Research in Computational Algebraen
dc.description.statusPeer revieweden

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