Conjugacy Class Graph of Some Non-Abelian Groups

Authors

  • Chinmayee Kumar Department of Mathematics, Gauhati University, India https://orcid.org/0000-0003-4490-8451
  • Kuntala Patra Department of Mathematics, Gauhati University, India

DOI:

https://doi.org/10.37256/cm.5120243875

Keywords:

non-abelian group, conjugacy class graph, line graph, energy (of a graph)

Abstract

The conjugacy class graph of a group G is a graph whose vertices are the non-central conjugacy classes of G and two vertices are adjacent if their cardinalities are not co-prime. In this paper, conjugacy class graphs of Dn, Q4n, Sn are studied. These graphs are found to be either complete graphs or union of complete graphs. Conjugacy classes of Dn × Dm are calculated and the results obtained are used to determine the structure of conjugacy class graphs of Dn × Dm, for odd and even values of m and n. Conjugacy class graphs of Dn are non-planar for n = 8 and n ≥ 11. They are non-hyperenergetic for all n and hypoenergetic only for n = 3, 5. Also, line graphs of these graphs are regular and eulerian for n ≡ 1 (mod 2) and n ≡ 0 (mod 4). The conjugacy class graphs of Q4n are non-planar for n = 4 and n ≥ 6. These graphs are non-hyperenergetic as well as non-hypoenergetic. The line graphs are eulerian for even values of n. It is conjectured that conjugacy class graph of Sn is non-planar for n ≥ 5.

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Published

2024-01-23

How to Cite

1.
Kumar C, Patra K. Conjugacy Class Graph of Some Non-Abelian Groups. Contemp. Math. [Internet]. 2024 Jan. 23 [cited 2024 Feb. 27];5(1):430-45. Available from: https://ojs.wiserpub.com/index.php/CM/article/view/3875