Cyclic Subgroup Graph of a Group

TL;DR

This paper investigates the properties of cyclic subgroup graphs of groups, focusing on their structure and characteristics, expanding understanding of subgroup relationships within group theory.

Contribution

It introduces new analyses of cyclic subgroup graphs, building on Mrnuceanu's edge-count formula to explore their properties and structure.

Findings

Characterization of cyclic subgroup graph properties

Relationships between subgroup structure and graph features

Extensions of previous edge-count results

Abstract

A cyclic subgroup graph of a group $G$ is a graph whose vertices are cyclic subgroups of $G$ and two distinct vertices $H_1$ and $H_2$ are adjacent if $H_1\leq H_2$, and there is no subgroup $K$ such that $H_1<K<H_2$. M.T\u{a}rn\u{a}uceanu gave the formula to count the number of edges of these graphs. In this paper, we explore various properties of these graphs.