The undirected power graph on the conjugacy classes of a finite group

Mahmood Robati, Sajjad

doi:10.22060/ajmc.2025.23715.1293

The undirected power graph on the conjugacy classes of a finite group

Document Type : Short Contribution

Author

Sajjad Mahmood Robati

Department of Pure Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin, Iran

10.22060/ajmc.2025.23715.1293

Abstract

Let $G$ be a finite group. The undirected power graph on the conjugacy classes of $G$ is the simple graph $\mathcal{P_C}(G)$ whose vertices are the conjugacy classes of $G$ and two distinct vertices $C$ and $C'$ are adjacent if one is a subset of a power of the other. In this paper, we show that the graph $\mathcal{P_C}(G)$ is $2$-connected whenever either $|\pi(G)|>1$ or ${\rm Z}(G)$ is cyclic. Moreover, we classify finite groups $G$ whose associated graph $\mathcal{P_C}(G)-\{e\}$ are bipartite.

Keywords

Conjugacy classes

$2$-Connected graph

Bipartite graph

Finite group

Subjects