On the Independence Number of the Prime-Coprime Graph of a Finite Group

TL;DR

This paper studies the prime-coprime graph of finite groups, characterizing when it is a split graph and calculating its independence number for various group families.

Contribution

It provides a complete characterization of groups with split prime-coprime graphs and explicit independence number calculations for key group classes.

Findings

Characterized all finite groups with split prime-coprime graphs.

Established a lower bound for the independence number of the graph.

Computed the independence number for cyclic, dihedral, dicyclic, and semidihedral groups.

Abstract

The prime-coprime graph $\Theta(G)$ of a finite group $G$ is the simple graph with vertex set $G$, where two distinct elements are adjacent whenever the greatest common divisor of their orders is either $1$ or a prime. We characterize all finite groups $G$ for which $\Theta(G)$ is a split graph. We establish a general lower bound for the independence number of $\Theta(G)$ of an arbitrary finite group $G$. Moreover, we explicitly compute the independence number of $\Theta(G)$ for several distinguished families of finite groups, including cyclic, dihedral, dicyclic, and semidihedral groups.