Forbidden Subgraphs of co-prime Graphs of finite Groups

TL;DR

This paper investigates the properties of finite groups based on the structure of their co-prime graphs, specifically identifying conditions that forbid certain subgraphs like cycles and stars.

Contribution

It characterizes finite groups whose co-prime graphs exclude specific subgraphs, advancing understanding of the relationship between group structure and graph properties.

Findings

Identifies groups with co-prime graphs that forbid $C_4$, $K_{1,3}$, $P_4$, and asteroidal triples.

Provides structural characterizations related to forbidden subgraphs.

Enhances the connection between group theory and graph theory.

Abstract

For a finite group $G$ the co-prime graph $\Gamma(G)$ is defined as a graph with vertex set $G$ in which two distinct vertices $x$ and $y$ are adjacent if and only if $gcd(o(x),o(y))=1$ where $o(x)$ and $o(y)$ denote the orders of the elements $x$ and $y$ respectively. In this paper we find properties of groups whose co-prime graphs forbid graphs such as $C_4,K_{1,3},P_4$ and asteroidal triples.