Open Access
2016 On the divisibility graph for finite sets of positive integers
Adeleh Abdolghafourian, Mohammad A. Iranmanesh
Rocky Mountain J. Math. 46(6): 1755-1770 (2016). DOI: 10.1216/RMJ-2016-46-6-1755


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.


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Adeleh Abdolghafourian. Mohammad A. Iranmanesh. "On the divisibility graph for finite sets of positive integers." Rocky Mountain J. Math. 46 (6) 1755 - 1770, 2016.


Published: 2016
First available in Project Euclid: 4 January 2017

zbMATH: 1352.05083
MathSciNet: MR3591260
Digital Object Identifier: 10.1216/RMJ-2016-46-6-1755

Primary: 05C25

Keywords: Bipartite graph , common divisor graph , connected component , Divisibility graph , prime vertex graph

Rights: Copyright © 2016 Rocky Mountain Mathematics Consortium

Vol.46 • No. 6 • 2016
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