Classification of Groups according to the number of end vertices in the coprime graph

TL;DR

This paper characterizes finite groups based on the number of end vertices in their coprime graphs, establishing bounds and classifications for groups with specific end vertex counts.

Contribution

It provides new characterizations and classifications of groups according to the number of end vertices in their coprime graphs, including bounds and unique group identifications.

Findings

2-groups are the only groups with an odd number of end vertices

Z4 and Z2×Z2 are the only groups with exactly three end vertices

An upper bound on group order depending on end vertices

Abstract

In this paper we characterize groups according to the number of end vertices in the associated coprime graphs. An upper bound on the order of the group that depends on the number of end vertices is obtained. We also prove that $2-$groups are the only groups whose coprime graphs have odd number of end vertices. Classifications of groups with small number of end vertices in the coprime graphs are given. One of the results shows that $\mathbb{Z}_4$ and $\mathbb{Z}_2\times \mathbb{Z}_2$ are the only groups whose coprime graph has exactly three end vertices.