Forbidden subgraphs on conjugacy class graphs of groups

Abstract

Let $G$ be a finite group. The \textit{commuting/nilpotent/solvable conjugacy class graph} ($\Gamma_{CCC}(G)$, $\Gamma_{NCC}(G)$, or $\Gamma_{SCC}(G)$) is a simple graph whose vertex set consists of all non-central conjugacy classes of $G$. Two vertices $x^G$ and $y^G$ are adjacent if and only if there exist elements $a \in x^G$ and $b \in y^G$ such that $\langle a, b \rangle$ forms an abelian, nilpotent, or solvable subgroup of $G$, respectively.\par In this paper, we mainly investigate cographs (it is $P_4$-free), chordal graphs (it is $C_n$-free $\forall \ n\ge 4$ ), split graphs (it contains no induced subgraph isomorphic to $C_4,\ C_5$, and $2K_2$), threshold graphs (it contains no induced subgraph isomorphic to $P_4$, $C_4,\ C_5$, and $2K_2$), and claw-free graphs (it contains no vertex with three pairwise non-adjacent neighbours) in terms of forbidden induced subgraphs in $\Gamma_{CCC}(G)$/ $\Gamma_{NCC}(G)$/$\Gamma_{SCC}(G)$.\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 $M_{11}$, $M_{12}$, and $M_{22}$, as well as certain minimal simple groups such as Suzuki groups and $\mathrm{PSL}(3,3)$. For other minimal simple groups, such as $\mathrm{PSL}(2,2^p)$, $\mathrm{PSL}(2,3^p)$, and $\mathrm{PSL}(2,p)$ (where $p > 3$ and $5 \mid p^2 + 1$), 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.