Commuting conjugacy classes graph of the generalized dihedral and dicyclic groups

Abstract

Suppose $G$ is a finite non-abelian group and $\Gamma(G)$ is a simple graph with the non-central conjugacy classes of $G$ as its vertex set. Two different non-central conjugacy classes $A$ and $B$ are assumed to be adjacent if and only if there are elements $a,b \in G$ such that $a \in A$, $b \in B$ and $ab = ba$. This graph is called the commuting conjugacy class graph of G. In this paper, the structure of the commuting conjugacy class graph of the generalized dihedral group $D_{(m,n)}$ and the generalized dicyclic group $Dic (A, y, x)$ are completely determined.

Keywords

• conjugacy classes
• commuting conjugacy classes graph
• the generalized dihedral group
• the generalized dicyclic groups