Undirected power graphs of semigroups

Chakrabarty, Ivy; Ghosh, Susmita; Sen, M. K.

doi:10.1007/s00233-008-9132-y

Cited by 258 publications

(230 citation statements)

References 4 publications

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“…There are many papers on assigning a graph to a group and algebraic properties of group by using the associated graph; for instance, see [1][2][3][4].…”

Section: Introduction and Resultsmentioning

confidence: 99%

The Cyclic Graph of a Finite Group

Long¹,

Wei

Guo³

2013

Algebra

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The cyclic graph Γ of a finite group is as follows: take as the vertices of Γ and join two distinct vertices and if ⟨ , ⟩ is cyclic. In this paper, we investigate how the graph theoretical properties of Γ affect the group theoretical properties of . First, we consider some properties of Γ and characterize certain finite groups whose cyclic graphs have some properties. Then, we present some properties of the cyclic graphs of the dihedral groups 2 and the generalized quaternion groups 4 for some . Finally, we present some parameters about the cyclic graphs of finite noncyclic groups of order up to 14.

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“…There are many papers on assigning a graph to a group and algebraic properties of group by using the associated graph; for instance, see [1][2][3][4].…”

Section: Introduction and Resultsmentioning

confidence: 99%

The Cyclic Graph of a Finite Group

Long¹,

Wei

Guo³

2013

Algebra

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“…The enhanced power graph G of a finite group G is the graph whose vertex set consists of G, in which two distinct vertices are adjacent if they generate a cyclic subgroup. The concepts of a power graph and an undirected power graph were first introduced by Kelarev and Quinn [18] and Chakrabarty et al [19], respectively. Since the paper deals only with undirected graphs, we use the term "power graph" to refer to an undirected power graph.…”

Section: Introductionmentioning

confidence: 99%

“…Since the paper deals only with undirected graphs, we use the term "power graph" to refer to an undirected power graph. In recent years, the study of power graphs has been growing (see [19][20][21][22][23][24][25][26][27][28][29][30]). Also, see [31] for a survey of results and open questions on power graphs.…”

Section: Introductionmentioning

confidence: 99%

Perfect codes in power graphs of finite groups

et al. 2017

Open Mathematics

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Abstract:The power graph of a finite group is the graph whose vertex set is the group, two distinct elements being adjacent if one is a power of the other. The enhanced power graph of a finite group is the graph whose vertex set consists of all elements of the group, in which two vertices are adjacent if they generate a cyclic subgroup. In this paper, we give a complete description of finite groups with enhanced power graphs admitting a perfect code. In addition, we describe all groups in the following two classes of finite groups: the class of groups with power graphs admitting a total perfect code, and the class of groups with enhanced power graphs admitting a total perfect code. Furthermore, we characterize several families of finite groups with power graphs admitting a perfect code, and several other families of finite groups with power graphs which do not admit perfect codes.

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“…Though his key goal was to address the issue of colouring, this initiated the formal study of exposing the relationship between algebra and graph theory and at advancing applications of one to the other. Till then, a lot of research, e.g., [11,2,3,1,8,6,7,4] has been done in connecting graph structures to various algebraic objects. Recently, intersection graphs associated with subspaces of vector spaces were studied in [10,12].…”

Section: Introductionmentioning

confidence: 99%

Nonzero Component Graph of a Finite Dimensional Vector Space

Das

2016

Communications in Algebra

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In this paper, we introduce a graph structure, called non-zero component graph on finite dimensional vector spaces. We show that the graph is connected and find its domination number and independence number. We also study the inter-relationship between vector space isomorphisms and graph isomorphisms and it is shown that two graphs are isomorphic if and only if the corresponding vector spaces are so. Finally, we determine the degree of each vertex in case the base field is finite.

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Undirected power graphs of semigroups

Cited by 258 publications

References 4 publications

The Cyclic Graph of a Finite Group

The Cyclic Graph of a Finite Group

Perfect codes in power graphs of finite groups

Nonzero Component Graph of a Finite Dimensional Vector Space

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