Conjugacy Class Graph of Some Non-Abelian Groups

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 D_n, Q_4n, S_n are studied. These graphs are found to be either complete graphs or union of complete graphs. Conjugacy classes of D_n × D_m are calculated and the results obtained are used to determine the structure of conjugacy class graphs of D_n × D_m, for odd and even values of m and n. Conjugacy class graphs of D_n 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 Q_4n 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 S_n is non-planar for n ≥ 5.