The Order Supergraph of the Power Graph of a Finite Group

TL;DR

This paper introduces the order supergraph of the power graph of a finite group, exploring its properties and relationship with the power graph to deepen understanding of group element interactions.

Contribution

It defines the order supergraph of the power graph and investigates its properties and connection to the power graph in finite groups.

Findings

Characterization of the order supergraph's properties

Relationship between power graph and order supergraph

Structural insights into group element divisibility

Abstract

The power graph $\mathcal{P}(G)$ is a graph with group elements as vertex set and two elements are adjacent if one is a power of the other. The order supergraph $\mathcal{S}(G)$ of the power graph $\mathcal{P}(G)$ is a graph with vertex set $G$ in which two elements $x, y \in G$ are joined if $o(x) | o(y)$ or $o(y) | o(x)$. The purpose of this paper is to study certain properties of this new graph together with the relationship between $\mathcal{P}(G)$ and $\mathcal{S}(G)$.