scholarly journals On the power graph of a finite group

Filomat ◽  
2012 ◽  
Vol 26 (6) ◽  
pp. 1201-1208 ◽  
Author(s):  
M. Mirzargar ◽  
A.R. Ashrafi ◽  
M.J. Nadjafi-Arani
Keyword(s):  
Finite Group ◽  
The Other ◽  
Power Graph ◽  
Vertex Set ◽  
A Finite Group

The power graph P(G) of a group G is the graph whose vertex set is the group elements and two elements are adjacent if one is a power of the other. In this paper, we consider some graph theoretical properties of a power graph P(G) that can be related to its group theoretical properties. As consequences of our results, simple proofs for some earlier results are presented.

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].

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.

scholarly journals Perfect codes in power graphs of finite groups

2017 ◽  
Vol 15 (1) ◽  
pp. 1440-1449 ◽  
Author(s):  
Xuanlong Ma ◽  
Ruiqin Fu ◽  
Xuefei Lu ◽  
Mengxia Guo ◽  
Zhiqin Zhao
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Cyclic Subgroup ◽  
The Other ◽  
Perfect Code ◽  
Perfect Codes ◽  
Power Graph ◽  
Vertex Set ◽  
A Finite Group

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.

Some properties of power graphs in finite group

2016 ◽  
Vol 09 (04) ◽  
pp. 1650079
Author(s):  
S. H. Jafari
Keyword(s):  
Finite Group ◽  
Group Theory ◽  
Finite Groups ◽  
The Other ◽  
Power Graph ◽  
Vertex Set ◽  
Group Ii ◽  
A Finite Group

The power graph of a group is the graph whose vertex set is the set of nontrivial elements of group, two elements being adjacent if one is a power of the other. We prove some beautiful results in power graphs of finite groups. Then we conclude two finite groups with isomorphic power graphs have the same number of elements of each order from the different way of [P. J. Cameron, The power graph of a finite group II, J. Group Theory 13 (2010) 779–783].

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.

scholarly journals On the characteristic polynomial of power graphs

Filomat ◽  
2018 ◽  
Vol 32 (12) ◽  
pp. 4375-4387
Author(s):  
Modjtaba Ghorbani ◽  
Fatemeh Abbasi-Barfaraz
Keyword(s):  
Finite Group ◽  
Characteristic Polynomial ◽  
The Other ◽  
Power Graph ◽  
Graphs Of Groups ◽  
Vertex Set

The power graph P(G) of finite group G is a graph whose vertex set is G and two distinct vertices are adjacent if one is a power of the other. In this paper, we determine the characteristic polynomial of the power graphs of groups of order a product of three primes.

Graceful labeling of power graph of group Z2k−1 × Z4

2021 ◽  
pp. 2250121
Author(s):  
Amit Sehgal ◽  
Neeraj Takshak ◽  
Pradeep Maan ◽  
Archana Malik
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Finite Abelian Group ◽  
Simple Graph ◽  
Graceful Labeling ◽  
Power Graph ◽  
Vertex Set ◽  
A Finite Group

The power graph of a finite group G is a special type of undirected simple graph whose vertex set is set of elements of G, in which two distinct vertices of G are adjacent if one is the power of other. Let [Formula: see text] be a finite abelian 2-group of order [Formula: see text] where [Formula: see text]. In this paper, we establish that the power graph of finite abelian group G always has graceful labeling without any condition on [Formula: see text].

The metric dimension of the enhanced power graph of a finite group

2019 ◽  
Vol 19 (01) ◽  
pp. 2050020 ◽  
Author(s):  
Xuanlong Ma ◽  
Yanhong She
Keyword(s):  
Finite Group ◽  
Dihedral Group ◽  
Metric Dimension ◽  
Quaternion Group ◽  
Power Graph ◽  
Vertex Set ◽  
Group A ◽  
Generalized Quaternion Group ◽  
A Finite Group ◽  
Generalized Quaternion

The enhanced power graph of a finite group [Formula: see text] is the graph whose vertex set is [Formula: see text], and two distinct vertices are adjacent if they generate a cyclic subgroup of [Formula: see text]. In this paper, we establish an explicit formula for the metric dimension of an enhanced power graph. As an application, we compute the metric dimension of the enhanced power graph of an elementary abelian [Formula: see text]-group, a dihedral group and a generalized quaternion group.

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.

Sign in / Sign up

Export Citation Format

Share Document