Properties of divisor prime graph

Abstract

Number theory is a mathematical discipline that uses concepts from graph theory.} Recently, various graphs have been defined in relation to various number theoretic functions. One such graph is the divisor prime graph, which is associated with the positive divisors of a positive integer. Let $n$ be a positive integer and $D(n)$ be the set of all positive divisors of $n$. The {\it divisor prime graph} $PG_D(n)$ is defined as a graph whose vertex set is $D(n)$ and any two vertices $x$ and $y$ are adjacent in $PG_D(n)$ iff $\gcd(x, y)=1$. In this study, families of divisor prime graphs for different positive integers are investigated, along with their graph theoretic characteristics such as adjacency, diameter, radius, clique number, chromatic number, planarity, connectivity, { independence number and density.