Some Properties of Order-Divisor Graphs of Finite Groups

TL;DR

This paper explores the structural properties of order-divisor graphs derived from finite groups, focusing on girth, vertex degree, and graph size, especially for cyclic and dihedral groups.

Contribution

It provides new detailed characterizations of order-divisor graphs for finite cyclic and dihedral groups, enhancing understanding of their graph-theoretic properties.

Findings

Determined girth, degree, and size for cyclic groups' order-divisor graphs.

Analyzed properties of dihedral groups' order-divisor graphs.

Established foundational results linking group structure to graph parameters.

Abstract

This article investigates the properties of order-divisor graphs associated with finite groups. An order-divisor graph of a finite group is an undirected graph in which the set of vertices includes all elements of the group, and two distinct vertices with different orders are adjacent if the order of one vertex divides the order of the other. We prove some beautiful results in order-divisor graphs of finite groups. The primary focus is on examining the girth, degree of vertices, and size of the order-divisor graph. In particular, we provide a comprehensive description of these parameters for the order-divisor graphs of finite cyclic groups and dihedral groups.