Forbidden subgraphs on conjugacy class graphs of groups

Abstract

Let $G$ be a finite group. The commuting (resp. nilpotent) conjugacy class graph $\Gamma_{CCC}(G)$ (resp. $\Gamma_{NCC}(G)$) of $G$ is a simple graph whose vertex set consists of all non-central conjugacy classes of $G$, in which two distinct vertices $x^G$ and $y^G$ are adjacent if and only if there exist $a \in x^G$ and $b \in y^G$ such that $\langle a, b \rangle$ is an abelian (resp. nilpotent) subgroup. In this paper, we mainly investigate cographs, chordal graphs, split graphs, threshold graphs, and claw-free graphs in terms of forbidden induced subgraphs in $\Gamma_{CCC}(G)$ and $\Gamma_{NCC}(G)$. To be specific, we characterize the induced subgraphs in the commuting conjugacy class graph for symmetric groups, alternating groups, and sporadic groups. We also provide a complete classification of these properties for EPPO-groups, nilpotent groups, dihedral groups, dicyclic groups, and generalized dihedral groups in both commuting and nilpotent conjugacy class groups.