Properties of divisor prime graph
Keywords:
Divisor, Prime factor, Greatest common divisor, Connectedness, Diameter, Girth, Radius, Isomorphism, Planar 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.
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