2024/06/03 by Ray, Papi, Arora, Sonakshee
#05C25 #20D15 #20E32 #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2406.01305
Let G be a finite group. The commuting/nilpotent/solvable conjugacy class graph (ΓCCC(G), ΓNCC(G), or ΓSCC(G)) is a simple graph whose vertex set consists of all non-central conjugacy classes of G. Two vertices xG and yG are adjacent if and only if there exist elements a ∈ xG and b ∈ yG such that ⟨ a, b ⟩ forms an abelian, nilpotent, or solvable subgroup of G, respectively.\par In this paper, we mainly investigate cographs (it is P4-free), chordal graphs (it is Cn-free ∀ n≥ 4 ), split graphs (it contains no induced subgraph isomorphic to C4, C5, and 2K2), threshold graphs (it contains no induced subgraph isomorphic to P4, C4, C5, and 2K2), and claw-free graphs (it contains no vertex with three pairwise non-adjacent neighbours) in terms of forbidden induced subgraphs in ΓCCC(G)/ ΓNCC(G)/ΓSCC(G).\par We provide a complete classification of these properties for EPPO groups, groups of order pq, and nilpotent groups. Additionally, we characterize the induced subgraphs in the commuting conjugacy class graph for symmetric and alternating groups. For solvable groups such as dihedral, dicyclic, and generalized dihedral groups, we establish complete results. Moreover, we fully characterize the graphs for the Mathieu groups M11, M12, and M22, as well as certain minimal simple groups such as Suzuki groups and PSL(3,3). For other minimal simple groups, such as PSL(2,2p), PSL(2,3p), and PSL(2,p) (where p > 3 and 5 | p2 + 1), we demonstrate that the solvable conjugacy class graph is always a cograph. Finally, we present several open problems, highlighting further directions for research in this area.