scholarly journals On finite groups whose power graph is a cograph

2021 ◽  
Author(s):  
Peter J. Cameron ◽  
Pallabi Manna ◽  
Ranjit Mehatari
Keyword(s):  
Finite Groups ◽  
Power Graph

scholarly journals On connectedness of power graphs of finite groups

2018 ◽  
Vol 17 (10) ◽  
pp. 1850184 ◽  
Author(s):  
Ramesh Prasad Panda ◽  
K. V. Krishna
Keyword(s):  
Finite Groups ◽  
Upper Bound ◽  
The Other ◽  
Number Of Components ◽  
Cyclic Groups ◽  
Power Graph ◽  
Vertex Set ◽  
Proper Power

The power graph of a group [Formula: see text] is the graph whose vertex set is [Formula: see text] and two distinct vertices are adjacent if one is a power of the other. This paper investigates the minimal separating sets of power graphs of finite groups. For power graphs of finite cyclic groups, certain minimal separating sets are obtained. Consequently, a sharp upper bound for their connectivity is supplied. Further, the components of proper power graphs of [Formula: see text]-groups are studied. In particular, the number of components of that of abelian [Formula: see text]-groups are determined.

On enhanced power graphs of finite groups

2018 ◽  
Vol 17 (08) ◽  
pp. 1850146 ◽  
Author(s):  
Sudip Bera ◽  
A. K. Bhuniya
Keyword(s):  
Finite Groups ◽  
Abelian Groups ◽  
Cyclic Subgroups ◽  
Power Graph ◽  
Vertex Set ◽  
One To One

Given a group [Formula: see text], the enhanced power graph of [Formula: see text], denoted by [Formula: see text], is the graph with vertex set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are edge connected in [Formula: see text] if there exists [Formula: see text] such that [Formula: see text] and [Formula: see text] for some [Formula: see text]. Here, we show that the graph [Formula: see text] is complete if and only if [Formula: see text] is cyclic; and [Formula: see text] is Eulerian if and only if [Formula: see text] is odd. We characterize all abelian groups and all non-abelian [Formula: see text]-groups [Formula: see text] such that [Formula: see text] is dominatable. Besides, we show that there is a one-to-one correspondence between the maximal cliques in [Formula: see text] and the maximal cyclic subgroups of [Formula: see text].

scholarly journals Finite groups whose intersection power graphs are toroidal and projective-planar

2021 ◽  
Vol 19 (1) ◽  
pp. 850-862
Author(s):  
Huani Li ◽  
Xuanlong Ma ◽  
Ruiqin Fu
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Identity Element ◽  
Power Graph ◽  
Vertex Set ◽  
A Finite Group

Abstract The intersection power graph of a finite group G G is the graph whose vertex set is G G , and two distinct vertices x x and y y are adjacent if either one of x x and y y is the identity element of G G , or ⟨ x ⟩ ∩ ⟨ y ⟩ \langle x\rangle \cap \langle y\rangle is non-trivial. In this paper, we completely classify all finite groups whose intersection power graphs are toroidal and projective-planar.

A study of enhanced power graphs of finite groups

2019 ◽  
Vol 19 (04) ◽  
pp. 2050062 ◽  
Author(s):  
Samir Zahirović ◽  
Ivica Bošnjak ◽  
Rozália Madarász
Keyword(s):  
Finite Groups ◽  
Prime Power ◽  
Cyclic Subgroup ◽  
Nilpotent Groups ◽  
Sufficient Condition ◽  
Finite Abelian Groups ◽  
Power Graph ◽  
Vertex Set ◽  
A Finite Group

The enhanced power graph [Formula: see text] of a group [Formula: see text] is the graph with vertex set [Formula: see text] such that two vertices [Formula: see text] and [Formula: see text] are adjacent if they are contained in the same cyclic subgroup. We prove that finite groups with isomorphic enhanced power graphs have isomorphic directed power graphs. We show that any isomorphism between undirected power graph of finite groups is an isomorphism between enhanced power graphs of these groups, and we find all finite groups [Formula: see text] for which [Formula: see text] is abelian, all finite groups [Formula: see text] with [Formula: see text] being prime power, and all finite groups [Formula: see text] with [Formula: see text] being square-free. Also, we describe enhanced power graphs of finite abelian groups. Finally, we give a characterization of finite nilpotent groups whose enhanced power graphs are perfect, and we present a sufficient condition for a finite group to have weakly perfect enhanced power graph.

Proper connection of power graphs of finite groups

2020 ◽  
pp. 2150033
Author(s):  
Xuanlong Ma
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Undirected Graph ◽  
The Other ◽  
Connection Number ◽  
Power Graph ◽  
Vertex Set ◽  
Proper Connection Number ◽  
A Finite Group

Let [Formula: see text] be a finite group. The power graph of [Formula: see text] is the undirected graph whose vertex set is [Formula: see text], and two distinct vertices are adjacent if one is a power of the other. The reduced power graph of [Formula: see text] is the subgraph of the power graph of [Formula: see text] obtained by deleting all edges [Formula: see text] with [Formula: see text], where [Formula: see text] and [Formula: see text] are two distinct elements of [Formula: see text]. In this paper, we determine the proper connection number of the reduced power graph of [Formula: see text]. As an application, we also determine the proper connection number of the power graph of [Formula: see text].

scholarly journals On the minimum degree of the power graph of a finite cyclic group

2020 ◽  
pp. 2150044
Author(s):  
Ramesh Prasad Panda ◽  
Kamal Lochan Patra ◽  
Binod Kumar Sahoo
Keyword(s):  
Finite Group ◽  
Cyclic Group ◽  
Finite Groups ◽  
Minimum Degree ◽  
Simple Graph ◽  
The Other ◽  
Prime Divisors ◽  
Power Graph ◽  
Vertex Set ◽  
A Finite Group

The power graph [Formula: see text] of a finite group [Formula: see text] is the undirected simple graph whose vertex set is [Formula: see text], in which two distinct vertices are adjacent if one of them is an integral power of the other. For an integer [Formula: see text], let [Formula: see text] denote the cyclic group of order [Formula: see text] and let [Formula: see text] be the number of distinct prime divisors of [Formula: see text]. The minimum degree [Formula: see text] of [Formula: see text] is known for [Formula: see text], see [R. P. Panda and K. V. Krishna, On the minimum degree, edge-connectivity and connectivity of power graphs of finite groups, Comm. Algebra 46(7) (2018) 3182–3197]. For [Formula: see text], under certain conditions involving the prime divisors of [Formula: see text], we identify at most [Formula: see text] vertices such that [Formula: see text] is equal to the degree of at least one of these vertices. If [Formula: see text], or that [Formula: see text] is a product of distinct primes, we are able to identify two such vertices without any condition on the prime divisors of [Formula: see text].

On spectra and spectral radius of Signless Laplacian of power graphs of some finite groups

2020 ◽  
pp. 2150090
Author(s):  
Subarsha Banerjee ◽  
Avishek Adhikari
Keyword(s):  
Spectral Radius ◽  
Finite Groups ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Laplacian Spectral Radius ◽  
Power Graph ◽  
Signless Laplacian Spectral Radius ◽  
Signless Laplacian ◽  
Extremal Values ◽  
A Finite Group

Let [Formula: see text] denote the power graph of a finite group [Formula: see text]. Let [Formula: see text] denote the Signless Laplacian spectral radius of [Formula: see text]. In this paper, we give lower and upper bounds on [Formula: see text] for any [Formula: see text] and find those graphs for which the extremal values are attained. We give a comparison between the bounds obtained and exact value of [Formula: see text] for any [Formula: see text]. We then find the eigenvalues of [Formula: see text] and give lower and upper bounds on the spectral radius of [Formula: see text]. When [Formula: see text] and [Formula: see text] where [Formula: see text] are primes and [Formula: see text] is a positive integer, we obtain sharper bounds on [Formula: see text]. Finally, we make a conjecture on [Formula: see text] for any [Formula: see text].

Normal subgroup based power graph of finite groups

2021 ◽  
Author(s):  
Nasrin Malek-Mohammadi ◽  
Ali Reza Ashrafi
Keyword(s):  
Normal Subgroup ◽  
Finite Groups ◽  
Power Graph

scholarly journals Spectrum and L-spectrum of the power graph and its main supergraph for certain finite groups

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5323-5334 ◽  
Author(s):  
Asma Hamzeh ◽  
Ali Ashrafi
Keyword(s):  
Finite Group ◽  
Characteristic Polynomial ◽  
Finite Groups ◽  
The Other ◽  
Laplacian Spectrum ◽  
Simple Graphs ◽  
Power Graph ◽  
Vertex Set ◽  
A Finite Group

Let G be a finite group. The power graph P(G) and its main supergraph S(G) are two simple graphs with the same vertex set G. Two elements x,y ? G are adjacent in the power graph if and only if one is a power of the other. They are joined in S(G) if and only if o(x)|o(y) or o(y)|o(x). The aim of this paper is to compute the characteristic polynomial of these graph for certain finite groups. As a consequence, the spectrum and Laplacian spectrum of these graphs for dihedral, semi-dihedral, cyclic and dicyclic groups were computed.

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