TY - JOUR
AU - Kumar, Chinmayee
AU - Patra, Kuntala
PY - 2024/01/23
Y2 - 2024/04/15
TI - Conjugacy Class Graph of Some Non-Abelian Groups
JF - Contemporary Mathematics
JA - Contemp. Math.
VL - 5
IS - 1
SE - Research Article
DO - 10.37256/cm.5120243875
UR - https://ojs.wiserpub.com/index.php/CM/article/view/3875
SP - 430-445
AB - <p>The conjugacy class graph of a group <em>G</em> is a graph whose vertices are the non-central conjugacy classes of<em> G</em> and two vertices are adjacent if their cardinalities are not co-prime. In this paper, conjugacy class graphs of <em>D<sub>n</sub></em>, <em>Q<sub>4n</sub></em>, <em>S<sub>n</sub></em> are studied. These graphs are found to be either complete graphs or union of complete graphs. Conjugacy classes of <em>D<sub>n</sub></em> × <em>D<sub>m</sub></em> are calculated and the results obtained are used to determine the structure of conjugacy class graphs of <em>D<sub>n</sub></em> × <em>D<sub>m</sub></em>, for odd and even values of <em>m</em> and <em>n</em>. Conjugacy class graphs of <em>D<sub>n</sub></em> are non-planar for <em>n</em> = 8 and <em>n</em> ≥ 11. They are non-hyperenergetic for all <em>n</em> and hypoenergetic only for <em>n</em> = 3, 5. Also, line graphs of these graphs are regular and eulerian for <em>n</em> ≡ 1 (mod 2) and <em>n</em> ≡ 0 (mod 4). The conjugacy class graphs of <em>Q<sub>4n</sub></em> are non-planar for <em>n</em> = 4 and <em>n</em> ≥ 6. These graphs are non-hyperenergetic as well as non-hypoenergetic. The line graphs are eulerian for even values of <em>n</em>. It is conjectured that conjugacy class graph of <em>S<sub>n</sub></em> is non-planar for <em>n</em> ≥ 5.</p>
ER -