Common Divisor Graph Research Articles

On the Common Divisor Graph of the Product of Integer Multisets

On the Common Divisor Graph of the Product of Integer Multisets

Five-vertex common divisor character degree graphs of non-solvable groups

For a finite group G, the common divisor graph attached to G is a character degree graph whose vertices are the degrees of the non-linear irreducible complex characters of G, and distinct vertices a, b are adjacent when they have a non-trivial common divisor. In this article, we determine all graphs with five vertices that may occur as for a non-solvable group G.

Finite Groups Whose Common-Divisor Graph is Regular

AbstractLet G be a finite group, and write cd (G) for the set of degrees of irreducible characters of G. The common-divisor graph Γ(G) associated with G is the graph whose vertex set is cd (G)∖{1} and there is an edge between distinct vertices a and b, if (a, b) > 1. In this paper we prove that if Γ(G) is a k-regular graph for some k ⩾ 0, then for the solvable groups, either Γ(G) is a complete graph of order k + 1 or Γ(G) has two connected components which are complete of the same order and for the non-solvable groups, either k = 0 and cd(G) = cd(PSL2(2f)), where f ⩾ 2 or Γ(G) is a 4-regular graph with six vertices and cd(G) = cd(Alt7) or cd(Sym7).

Common Divisor Character Degree Graphs of Solvable Groups with Four Vertices

Let G be a finite solvable group. The common divisor graph Γ(G) attached to G is a character degree graph. Its vertices are the degrees of the nonlinear irreducible complex characters of G, and different vertices m, n are adjacent if the greatest common divisor (m, n) > 1. In this article, we classify all graphs with four vertices that may occur as Γ(G) for solvable group G.

On Graphs Associated with Character Degrees and Conjugacy Class Sizes of Direct Products of Finite Groups

Abstract.The prime vertex graph, Δ(X), and the common divisor graph, Γ(X), are two graphs that have been deûned on a set of positive integers X. Some properties of these graphs have been studied in the cases where either X is the set of character degrees of a group or X is the set of conjugacy class sizes of a group. In this paper, we gather some results on these graphs arising in the context of direct product of two groups.

Graphs attached to simple Frobenius-Perron dimensions of an integral fusion category

Let \({\mathcal C}\) be an integral fusion category. We study some graphs, called the prime graph and the common divisor graph, related to the Frobenius-Perron dimensions of simple objects in the category \({\mathcal C}\), that extend the corresponding graphs associated to the irreducible character degrees and the conjugacy class sizes of a finite group. We describe these graphs in several cases, among others, when \({\mathcal C}\) is an equivariantization under the action of a finite group, a \(2\)-step nilpotent fusion category, and the representation category of a twisted quantum double. We prove generalizations of known results on the number of connected components of the corresponding graphs for finite groups in the context of braided fusion categories. In particular, we show that if \({\mathcal C}\) is any integral non-degenerate braided fusion category, then the prime graph of \({\mathcal C}\) has at most \(3\) connected components, and it has at most \(2\) connected components if \({\mathcal C}\) is in addition solvable. As an application we prove a classification result for weakly integral braided fusion categories all of whose simple objects have prime power Frobenius-Perron dimension.

Bipartite Divisor Graphs for Integer Subsets

Inspired by connections described in a recent paper by Mark L. Lewis, between the common divisor graph Γ(X) and the prime vertex graph Δ(X), for a set X of positive integers, we define the bipartite divisor graph B(X), and show that many of these connections flow naturally from properties of B(X). In particular we establish links between parameters of these three graphs, such as number and diameter of components, and we characterise bipartite graphs that can arise as B(X) for some X. Also we obtain necessary and sufficient conditions, in terms of subconfigurations of B(X), for one of Γ(X) or Δ(X) to contain a complete subgraph of size 3 or 4.

On IP-graphs of association schemes and applications to group theory

Let G be a group acting transitively on a set X such that all subdegrees are finite. Isaacs and Praeger (1993) [5] studied the common divisor graph of ( G , X ) . For a group G and its subgroup A, based on the results in Isaacs and Praeger (1993) [5], Kaplan (1997) [6] proved that if A is stable in G and the common divisor graph of ( A , G ) has two components, then G has a nice structure. Motivated by the notion of the common divisor graph of ( G , X ) , Camina (2008) [3] introduced the concept of the IP-graph of a naturally valenced association scheme. The common divisor graph of ( G , X ) is the IP-graph of the association scheme arising from the action of G on X. Xu (2009) [8] studied the properties of the IP-graph of an arbitrary naturally valenced association scheme, and generalized the main results in Isaacs and Praeger (1993) [5] and Camina (2008) [3]. In this paper we first prove that if the IP-graph of a naturally valenced association scheme ( X , S ) is stable and has two components (not including the trivial component whose only vertex is 1), then S has a closed subset T such that the thin residue O ϑ ( T ) and the quotient scheme ( X / O ϑ ( T ) , S / / O ϑ ( T ) ) have very nice properties. Then for an association scheme ( X , S ) and a closed subset T of S such that S / / T is an association scheme on X / T , we study the relations between the closed subsets of S and those of S / / T . Applying these results to schurian schemes and common divisor graphs of groups, we obtain the results of Kaplan [6] as direct consequences.

Valencies and IP-graphs of association schemes

Let G be a group acting transitively on a set X such that all subdegrees are finite. Isaacs and Praeger [I.M. Isaacs, C.E. Praeger, Permutation group subdegrees and the common divisor graph, J. Algebra 159 (1993) 158–175] and Kaplan [G. Kaplan, On groups admitting a disconnected common divisor graph, J. Algebra 193 (1997) 616–628] studied the common divisor graph of ( G , X ) . The action of G on X induces an association scheme ( X , S ) . Recently, motivated by the common divisor graph of ( G , X ) , Camina [R. Camina, Schemes and the IP-graph, J. Algebraic Combin. 28 (2008) 271–278] introduced the IP-graph of a naturally valenced association scheme. The common divisor graph of ( G , X ) is the IP-graph of the naturally valenced association scheme ( X , S ) arising from the action of G on X. Under a very strong assumption that all paired valencies are equal, Camina [R. Camina, Schemes and the IP-graph, J. Algebraic Combin. 28 (2008) 271–278] proved that the main results in [I.M. Isaacs, C.E. Praeger, Permutation group subdegrees and the common divisor graph, J. Algebra 159 (1993) 158–175] are also true for the IP-graph of a naturally valenced association scheme with paired valencies equal. However, the association scheme arising from the action of a group acting transitively on a set may not satisfy this assumption. The purpose of this paper is to prove similar results for IP-graphs of naturally valenced association schemes without the assumption that all paired valencies are equal, and hence generalize results in [I.M. Isaacs, C.E. Praeger, Permutation group subdegrees and the common divisor graph, J. Algebra 159 (1993) 158–175; R. Camina, Schemes and the IP-graph, J. Algebraic Combin. 28 (2008) 271–278; G. Kaplan, On groups admitting a disconnected common divisor graph, J. Algebra 193 (1997) 616–628].

Schemes and the IP-graph

We consider the common-divisor graph of the set of valencies of a naturally valenced scheme, where scheme is defined in the sense of P.-H. Zieschang. We prove structural results about this graph, and thus give restrictions on the set of natural numbers that can occur as the set of valencies of a naturally valenced scheme.

Strongly connective degree sets II: perfect derived subgroups

When G is a finite nonabelian group, we associate the common-divisor graph with G by letting nontrivial degrees in cd(G) = {χ(1) | χ∈Irr(G)} be the vertices and making distinct vertices adjacent if they have a common nontrivial divisor. A set \({\user1{\mathcal{C}}}\) of vertices for this graph is said to be strongly connective for cd(G) if there is some prime which divides every member of \({\user1{\mathcal{C}}}\), and every vertex outside of \({\user1{\mathcal{C}}}\) is adjacent to some member of \({\user1{\mathcal{C}}}\). When G is nonsolvable, we provide sufficiency conditions for cd(G) to have a strongly connective subset. We also extend a previously known result about groups with nonabelian solvable quotients, and prove for arbitrary groups G that if the associated graph is connected and has a diameter bounded by 2, then indeed cd(G) has a strongly connective subset. The major focus is on when the derived subgroup G′ is perfect.

Strongly connective degree sets I: nonabelian solvable quotients

When G is a finite nonabelian group, we associate the common-divisor graph Γ(G) with G by letting nontrivial degrees in cd(G) be the vertices and making distinct vertices adjacent if they have a common nontrivial divisor. A set \(\mathcal{C}\) of vertices for this graph is said to be strongly connective for cd(G) if there is some prime which divides every member of \(\mathcal{C},\) and every vertex outside of \(\mathcal{C}\) is adjacent to some member of \(\mathcal{C}.\) When G has a nonabelian solvable quotient, we show that if Γ(G) is connected and has a diameter of at most 2, then indeed cd(G) has a strongly connective subset.

Bounding graph diameters of nonsolvable groups

Associated with the character degrees of a finite group is the common-divisor graph, where the nontrivial degrees are the vertices and distinct degrees are adjacent when they have a common nontrivial divisor. The author has shown that common-divisor graphs for solvable groups have diameters of at most 3. The present paper extends this result to nonsolvable groups by showing that their common-divisor graphs also have diameters bounded above by 3.

Character degree graphs and normal subgroups

We consider the degrees of those irreducible characters of a group G G whose kernels do not contain a given normal subgroup N N . We show that if N ⊆ G ′ N \subseteq G’ and N N is not perfect, then the common-divisor graph on this set of integers has at most two connected components. Also, if N N is solvable, we obtain bounds on the diameters of the components of this graph and, in the disconnected case, we study the structure of N N and of G G .

Irreducible disconnected systems in groups

LetG be an arbitrary group with a subgroupA. The subdegrees of (A, G) are the indices [A:A ∪A9] (wheregeG). Equivalent definitions of that concept are given in [IP] and [K]. IfA is not normal inG and all the subdegrees of (A, G) are finite, we attach to (A, G) the common divisor graph Γ: its vertices are the non-unit subdegrees of (A, G), and two different subdegrees are joined by an edge iff they arenot coprime. It is proved in [IP] that Γ has at most two connected components. Assume that Γ is disconnected. LetD denote the subdegree set of (A, G) and letD1 be the set of all the subdegrees in the component of Γ containing min(D−{1}). We proved [K, Theorem A] that ifA is stable inG (a property which holds whenA or [G:A] is finite), then the setH={g e G| [A:A ∪Ag] eD1 ∪ {1}} is a subgroup ofG. In this case we say thatA<H<G is a disconnected system (briefly: a system). In the current paper we deal with some fundamental types of systems. A systemA<H<G is irreducible if there does not exist 1<N△G such thatAN<H andAN/N<H/N<G/N is a system. Theorem A gives restrictions on the finite nilpotent normal subgroups ofG, whenG possesses an irreducible system. In particular, ifG is finite then Fit(G) is aq-group for a certain primeq. We deal also with general systems. Corollary (4.2) gives information about the structure of a finite groupG which possesses a system. Theorem B says that for any systemA<H<G,NG(NG(A))=NG(A). Theorem C and Corollary C’ generalize a result of Praeger [P, Theorem 2].

On Groups Admitting a Disconnected Common Divisor Graph

LetGbe an arbitrary group with a subgroupA. Each double cosetAgAis a union of right cosetsAu. The cardinality of the set {Au∣u∈G,Au⊆AgA} is called asubdegreeof (A,G) and is denoted by [AgA:A]. Thus for each double cosetAgAwe have a corresponding subdegree. An equivalent definition of the subdegree concept is given in [2]. IfAis not normal inGand all the subdegrees of (A,G) are finite, we attach to (A,G) thecommon divisor graphΓ: its vertices are the nonunit subdegrees of (A,G), and two different subdegrees are joined by an edge iff they arenotcoprime. It is proved in [2] that Γ has at most two connected components. We prove that if Γ is disconnected andAsatisfies a certain “regularity” property (a property which holds whenAor [G:A] is finite, and is called in this paper stability), thenGhas a nice structure. To be more precise, letDdenote the subdegree set of (A,G) and letD1be the set of all the subdegrees in the connected component of Γ containing min(D−{1}). Then we prove (Theorem A) that the setH=⋃[AgA:A]∈D1∪{1}AgAis a subgroup ofGandNG(A)<H<G. Some interesting properties of the subgroupHare described in Theorem B. Theorems D and E describe some properties of the subdegrees in the disconnected case.

Permutation Group Subdegrees and the Common Divisor Graph

Permutation Group Subdegrees and the Common Divisor Graph