Bipartite Divisor Graph for the Product of Subsets of Integers

Abstract

The bipartite divisor graph B(X), for a set X of positive integers, and some of its properties have recently been studied. We construct the bipartite divisor graph for the product of subsets of positive integers and investigate some of its properties. We also give some applications in group theory.2010 Mathematics subject classification: primary 05C25; secondary 05C75.

On regular bipartite divisor graph for the set of irreducible character degrees

“…We conclude this brief discussion on B(Cl(G)) recalling that the first author and Iranmanesh [12,Theorem 4.1] have classified the groups G where B(Cl(G)) is isomorphic to a path. This classification was obtained by investigating the combinatorial properties of the bipartite divisor graphs constructed from the product of subsets of positive integers [12].…”

“…and α 3 , where α 1 := (1, 2)(3, 14, 9, 20)(4, 15, 19, 27)(5, 7)(6, 8)(10, 21, 18, 26)(11, 22, 12, 23)(13, 16)(17, 31, 25, 29)(24, 32, 28, 30), α 2 := (2, 27, 25, 19, 21, 10, 31)(3,29,8,22,17,11,14)(4,20,9,30,16,15,24)(7,23,28,12,26,18,32),…”

“…Therefore B(G) is a path of length 6. Furthermore, considering G as a permutation group of degree 32, it is generated by α 1 , α 2 , and α 3 , whereα 1 := (1, 2)(3,14, 9, 20)(4, 15, 19, 27)(5, 7)(6, 8)(10, 21, 18, 26)(11, 22, 12, 23)(13, 16)(17, 31, 25, 29)(24, 32, 28, 30), α 2 := (2, 27, 25, 19, 21, 10, 31)(3, 29, 8, 22, 17, 11, 14)(4, 20, 9, 30, 16, 15, 24)(7,23,28,12,26,18,32), Now that we have established the existence of a group G with B(G) ∼ = P n for n ∈ {5, 6}, it would be interesting to give a classification of this family of groups.Problem 2.8. Give structural information on the finite groups G with B(G) ∼ = P n for n ∈ {5, 6}.…”

Groups having complete bipartite divisor graphs for their conjugacy class sizes

“…One of the main questions that naturally arises in this area is classifying the groups whose bipartite divisor graphs have special graphical shapes. For instance, in [9], the first author of this paper and Iranmanesh have classified the groups whose bipartite divisor graphs are paths. Similarly, Taeri [13] considered the case that the bipartite divisor graph is a cycle, and in the course of his investigation posed the following question: Question.…”

On divisibility graph for simple Zassenhaus groups

“…There are several graphs associated to algebraic structures, specially finite groups, and many interesting results have been obtained recently (see for example [4,8,9,10] ). In [3] a new graph namely divisibility graph which is related to a set of positive integers have been introduced.…”