On the divisibility graph for finite sets of positive integers

Abstract

Let $X$ be a finite set of positive integers. The divisibility graph $\mathscr {D}\,(X)$ is a directed graph with vertex set $X\backslash \{1\}$ and an arc from $a$ to $b$ whenever $a$ divides $b$. Since the divisibility graph and its underlying graph have the same number of connected components, we consider the underlying graph of $\mathscr {D}\,(X)$, and we denote it by $\rm D (X)$. In this paper, we will find some graph theoretical parameters of $\rm D (X)$, some relations between the structure of $\rm D (X)$, and the structure of known graphs $\Gamma (X)$, $\Delta (X)$ and $B(X)$ will be considered. In addition, we investigate some properties of $\rm D (XY)$ for the product of two non-empty subsets $X$ and $Y$ of positive integers.