On the divisibility graph for finite sets of positive integers

Abstract

The divisibility graph D (X) is a directed graph with vertex set X\{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 D (X), and we denote it by D(X). In this paper, we will find some graph theoretical parameters of D(X), some relations between the structure of D(X), and the structure of known graphs Γ(X), ∆(X) and B(X) will be considered. In addition, we investigate some properties of D(XY ) for the product of two non-empty subsets X and Y of positive integers. 2010 AMS Mathematics subject classification. Primary 05C25.

Patterns of primes and composites on divisibility graph

“…Praeger and Iranmanesh defined the bipartite divisor graph whose vertex set is a disjoint union of the vertex sets of the prime vertex graph and the common divisor graph [7]. In [8], the authors study a undirected version of the divisibility graph D(X), as a simple graph with vertex set X* (= X − {1}) and two elements of X* are adjacent if one of them divides the other. D(X) and D(X) contain multiple components.…”

The Divisibility Graph of finite groups of Lie type

“…The first and second authors have shown that for every comparability graph, there is a finite set X such that this graph is isomorphic to D(X) in [3]. They found some relationships between the combinatorial properties of D(X), Γ(X) and ∆(X) such as the number of connected components, diameter and girth (see [3,Lemma 1]) and found a relationship between D(X × Y ) and product of D(X) and D(Y ) in [3]. They examined the Divisibility Graph D(G) of a finite group G in [1] and showed that when G is the symmetric or alternating group, then D(G) has at most two or three connected components, respectively.…”

“…The first and second authors have shown that for every comparability graph, there is a finite set X such that this graph is isomorphic to D(X) in [3]. They found some relationships between the combinatorial properties of D(X), Γ(X) and ∆(X) such as the number of connected components, diameter and girth (see [3, Lemma 1]) and found a relationship between D(X × Y ) and product of D(X) and D(Y ) in [3].…”