Super graphs on groups, I

G. Arunkumar, Peter J. Cameron*, Rajat Kanti Nath, Lavanya Selvaganesh

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Let G be a finite group. A number of graphs with the vertex set G have been studied, including the power graph, enhanced power graph, and commuting graph. These graphs form a hierarchy under the inclusion of edge sets, and it is useful to study them together. In addition, several authors have considered modifying the definition of these graphs by choosing a natural equivalence relation on the group such as equality, conjugacy, or equal orders, and joining two elements if there are elements in their equivalence class that are adjacent in the original graph. In this way, we enlarge the hierarchy into a second dimension. Using the three graph types and three equivalence relations mentioned gives nine graphs, of which in general only two coincide; we find conditions on the group for some other pairs to be equal. These often define interesting classes of groups, such as EPPO groups, 2-Engel groups, and Dedekind groups.

We study some properties of graphs in this new hierarchy. In particular, we
characterize the groups for which the graphs are complete, and
in most cases, we characterize the dominant vertices (those joined to all
others). Also, we give some results about universality,  perfectness, and clique
Original languageEnglish
Article number100
Number of pages14
JournalGraphs and Combinatorics
Issue number3
Early online date23 May 2022
Publication statusPublished - Jun 2022


  • Power graph
  • Commuting graph
  • Conjugacy
  • 2-Engel groups
  • EPPO groups


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