A survey on conjugacy class graphs of groups

Peter J. Cameron, Ferdous Ee Jannat, Rajat Kanti Nath, Reza Sharafdini

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Abstract

There are several graphs defined on groups. Among them we consider graphs whose vertex set consists conjugacy classes of a group G and adjacency is defined by properties of the elements of conjugacy classes. In particular, we consider commuting/nilpotent/solvable conjugacy class graph of G where two distinct onjugacy classes aG and bG are adjacent if there exist some elements x ∈ aG and y ∈ bG such that ⟨x y⟩ is abelian/nilpotent/solvable. After a section of introductory results and examples, we discuss all the available results on connectedness, graph realization, genus, various spectra and energies of certain induced subgraphs of these graphs. Proofs of the results are not included. However, many open problems for further investigation are stated.
Original languageEnglish
Article number125585
JournalExpositiones Mathematicae
Volume42
Issue number4
Early online date11 Jun 2024
DOIs
Publication statusE-pub ahead of print - 11 Jun 2024

Keywords

  • Commuting/nilpotent/solvable conjugacy class graph
  • Connectedness
  • Genus
  • Spectrum and energy
  • Induced subgraph

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