Proper divisor graph of a positive integer

TL;DR

This paper investigates the properties of the proper divisor graph of a positive integer, analyzing its structure and parameters to understand its role in algebraic graph theory and zero divisor graphs.

Contribution

It provides a comprehensive study of the graph parameters and automorphism group of the proper divisor graph, a novel graph associated with positive integers.

Findings

Determined clique, chromatic, and independence numbers of the graph.

Calculated the automorphism group of the proper divisor graph.

Analyzed various covering and matching numbers.

Abstract

The proper divisor graph $\Upsilon_n$ of a positive integer $n$ is the simple graph whose vertices are the proper divisors of $n$, and in which two distinct vertices $u, v$ are adjacent if and only if $n$ divides $uv$. The graph $\Upsilon_n$ plays an important role in the study of the zero divisor graph of the ring $\mathbb{Z}_n$. In this paper, we study some graph theoretic properties of $\Upsilon_n$ and determine the graph parameters such as clique number, chromatic number, chromatic index, independence number, matching number, domination number, vertex and edge covering numbers of $\Upsilon_n$. We also determine the automorphism group of $\Upsilon_n$.