On the number of edges of cyclic subgroup graphs of finite groups

TL;DR

This paper investigates the structure of cyclic subgroup graphs of finite groups, establishing minimal edge counts for certain classes and conjecturing a general property for all finite groups.

Contribution

It proves that cyclic groups have the fewest edges in their subgroup graphs among nilpotent and odd-order groups, and conjectures this extends to all finite groups.

Findings

Cyclic groups minimize edges among nilpotent groups

Cyclic groups minimize edges among odd-order groups

Conjecture: this minimality holds for all finite groups

Abstract

In this note, we show that among finite nilpotent groups of a given order or finite groups of a given odd order, the cyclic group of that order has the minimum number of edges in its cyclic subgroup graph. We also conjecture that this holds for arbitrary finite groups.