Intersection graphs of cyclic subgroups of groups

Rajkumar, R.; Devi, P.

doi:10.1016/j.endm.2016.05.003

Cited by 13 publications

(6 citation statements)

References 11 publications

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“…Another idea to obtain graphs related to a finite group G is to use different subsets of L(G) as being the vertex set and define the adjacency relation in a specific way. Some examples would be the intersection graph of G (see [26,36,40,42,44]), the join graph of G (see [3,35]) or the recent factorization graph of G (see [24]).…”

Section: Introductionmentioning

confidence: 99%

A graph related to the sum of element orders of a finite group

Lazorec

2023

CDM

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A finite group is called $\psi$-divisible iff $\psi(H)|\psi(G)$ for any subgroup $H$ of a finite group $G$. Here, $\psi(G)$ is the sum of element orders of $G$. For now, the only known examples of such groups are the cyclic ones of square-free order. The existence of non-abelian $\psi$-divisible groups still constitutes an open question. The aim of this paper is to make a connection between the $\psi$-divisibility property and graph theory. Hence, for a finite group $G$, we introduce a simple undirected graph called the $\psi$-divisibility graph of $G$. We denote it by $\psi_G$. Its vertices are the non-trivial subgroups of $G$, while two distinct vertices $H$ and $K$ are adjacent iff $H\subset K$ and $\psi(H)|\psi(K)$ or $K\subset H$ and $\psi(K)|\psi(H)$. We prove that $G$ is $\psi$-divisible iff $\psi_G$ has a universal (dominating) vertex. Also, we study various properties of $\psi_G$, when $G$ is a finite cyclic group. The choice of restricting our study to this specific class of groups is motivated in the paper.

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Section: Introductionmentioning

confidence: 99%

A graph related to the sum of element orders of a finite group

Lazorec

2023

CDM

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“…Another idea to obtain graphs related to a finite group G is to use different subsets of L(G) as being the vertex set and define the adjacency relation in a specific way. Some examples would be the intersection graph of G (see [27,37,40,42,44]), the join graph of G (see [3,36]) or the recent factorization graph of G (see [25]).…”

Section: Introductionmentioning

confidence: 99%

A graph related to the sum of element orders of a finite group

Lazorec¹

2022

Preprint

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A finite group is called ψ-divisible iff ψ(H)|ψ(G) for any subgroup H of a finite group G. Here, ψ(G) is the sum of element orders of G. For now, the only known examples of such groups are the cyclic ones of square-free order. The existence of non-abelian ψ-divisible groups still constitutes an open question. The aim of this paper is to make a connection between the ψ-divisibility property and graph theory. Hence, for a finite group G, we introduce a simple undirected graph called the ψ-divisibility graph of G. We denote it by ψ G . Its vertices are the non-trivial subgroups of G, while two distinct vertices H and K are adjacent iff H ⊂ K and ψ(H)|ψ(K) or K ⊂ H and ψ(K)|ψ(H). We prove that G is ψ-divisible iff ψ G has a universal (dominating) vertex. Also, we study various properties of ψ G , when G is a finite cyclic group. The choice of restricting our study to this specific class of groups is motivated in the paper.

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“…Later in [7], Bertram used the combinatorial properties of the commuting graph to prove three fundamental and non-trivial theorems on finite groups. Recently, Rajkumar and Devi in [19], introduced intersection graph of cyclic subgroups of finite groups and classify finite groups whose intersection graphs of cyclic subgroups is one of totally disconnected, complete, star, path, cycle. Beside these graphs, power graph is another graph of groups that was introduced in [14] as a tool to studying the combinatorial properties of groups with infinite sequence.…”

Section: Introductionmentioning

confidence: 99%

Graph coloring using commuting order product prime graph

Bello¹,

Ali²,

Isah³

2020

J. Math. Computer Sci.

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The concept of graph coloring has become a very active field of research that enhances many practical applications and theoretical challenges. Various methods have been applied in carrying out this study. Let G be a finite group. In this paper, we introduce a new graph of groups, which is a commuting order product prime graph of finite groups as a graph having the elements of G as its vertices and two vertices are adjacent if and only if they commute and the product of their order is a prime power. This is an extension of the study for order product prime graph of finite groups. The graph's general presentations on dihedral groups, generalized quaternion groups, quasi-dihedral groups, and cyclic groups have been obtained in this paper. Moreover, the commuting order product prime graph on these groups has been classified as connected, complete, regular, or planar. These results are used in studying various and recently introduced chromatic numbers of graphs.

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Intersection graphs of cyclic subgroups of groups

Cited by 13 publications

References 11 publications

A graph related to the sum of element orders of a finite group

A graph related to the sum of element orders of a finite group

A graph related to the sum of element orders of a finite group

Graph coloring using commuting order product prime graph

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