On some structural properties of a divisor graph

¹ Department of Applied Mathematics, School of Engineering, Samarkand International University of Technology, Samarkand 140100, Uzbekistan
² Department of School Education, JK Govt., Kashmir, India

^* Corresponding author: hilahmad1119kt@gmail.com

Received: 5 September 2024
Accepted: 21 March 2025

Abstract

A divisor graph G_n of a positive integer n is a simple graph with vertices as proper divisors of n, in which two distinct divisors are adjacent if and only if they are relatively prime. Also G′_n is a graph with vertices as divisors of n except n, such that the two distinct vertices are adjacent if and only if their greatest common divisor is one. These graph play a fundamental role in the study of the comaximal graphs associated to commutative rings. We study the graph theoretic properties of both G_n and G′_n. Formally, we determine the clique number, the independence number, the chromatic number and the automorphism group of the divisor graphs G_n and G′_n. We also find the bounds for their metric dimension along with the characterization of graphs attaining them. Further, we discuss their resolving polynomial.