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.

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.

On the structure of finite groups with dominatable enhanced power graph

2021 ◽  
pp. 2250176
Author(s):  
A. Mahmoudifar ◽  
A. Babai
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Open Problem ◽  
Identity Element ◽  
Quaternion Group ◽  
Power Graph ◽  
Vertex Set ◽  
Number Formula ◽  
Generalized Quaternion Group ◽  
Generalized Quaternion

Let [Formula: see text] be a group. The enhanced power graph of [Formula: see text] is a graph with vertex set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if there exists [Formula: see text] such that [Formula: see text] and [Formula: see text] for some [Formula: see text]. Also, a vertex of a graph is called dominating vertex if it is adjacent to every other vertex of the vertex set. Moreover, an enhanced power graph is said to be a dominatable graph if it has a dominating vertex other than the identity element. In an article of 2018, Bera and his coauthor characterized all abelian finite groups and nonabelian finite [Formula: see text]-groups such that their enhanced power graphs are dominatable (see [2]). In addition as an open problem, they suggested characterizing all finite nonabelian groups such that their enhanced power graphs are dominatable. In this paper, we try to answer their question. We prove that the enhanced power graph of finite group [Formula: see text] is dominatable if and only if there is a prime number [Formula: see text] such that [Formula: see text] and the Sylow [Formula: see text]-subgroups of [Formula: see text] are isomorphic to either a cyclic group or a generalized quaternion group.

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.

scholarly journals Certain properties of the power graph associated with a finite group

2014 ◽  
Vol 13 (07) ◽  
pp. 1450040 ◽  
Author(s):  
A. R. Moghaddamfar ◽  
S. Rahbariyan ◽  
W. J. Shi
Keyword(s):  
Finite Groups ◽  
Identity Element ◽  
Sufficient Conditions ◽  
Strongly Regular Graph ◽  
Simple Graph ◽  
Power Graph ◽  
Vertex Set ◽  
Strongly Regular ◽  
A Finite Group ◽  
Necessary And Sufficient

The power graph [Formula: see text] of a group G is a simple graph whose vertex-set is G and two vertices x and y in G are adjacent if and only if one of them is a power of the other. The subgraph [Formula: see text] of [Formula: see text] is obtained by deleting the vertex 1 (the identity element of G). In this paper, we first investigate some properties of the power graph [Formula: see text] and its subgraph [Formula: see text]. We next provide necessary and sufficient conditions for a power graph [Formula: see text] to be a strongly regular graph, a bipartite graph or a planar graph. Finally, we obtain some infinite families of finite groups G for which the power graph [Formula: see text] contains some cut-edges.

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 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.

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.

scholarly journals Non-Abelian finite groups whose character sums are invariant but are not Cayley isomorphism

2019 ◽  
Vol 18 (01) ◽  
pp. 1950013
Author(s):  
Alireza Abdollahi ◽  
Maysam Zallaghi
Keyword(s):  
Finite Group ◽  
Cayley Graph ◽  
Finite Groups ◽  
Closed Subset ◽  
Cayley Graphs ◽  
Character Sums ◽  
Main Question ◽  
Vertex Set ◽  
A Finite Group ◽  
Positive Integers

Let [Formula: see text] be a group and [Formula: see text] an inverse closed subset of [Formula: see text]. By a Cayley graph [Formula: see text], we mean the graph whose vertex set is the set of elements of [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] are adjacent if [Formula: see text]. A group [Formula: see text] is called a CI-group if [Formula: see text] for some inverse closed subsets [Formula: see text] and [Formula: see text] of [Formula: see text], then [Formula: see text] for some automorphism [Formula: see text] of [Formula: see text]. A finite group [Formula: see text] is called a BI-group if [Formula: see text] for some inverse closed subsets [Formula: see text] and [Formula: see text] of [Formula: see text], then [Formula: see text] for all positive integers [Formula: see text], where [Formula: see text] denotes the set [Formula: see text]. It was asked by László Babai [Spectra of Cayley graphs, J. Combin. Theory Ser. B 27 (1979) 180–189] if every finite group is a BI-group; various examples of finite non-BI-groups are presented in [A. Abdollahi and M. Zallaghi, Character sums of Cayley graph, Comm. Algebra 43(12) (2015) 5159–5167]. It is noted in the latter paper that every finite CI-group is a BI-group and all abelian finite groups are BI-groups. However, it is known that there are finite abelian non-CI-groups. Existence of a finite non-abelian BI-group which is not a CI-group is the main question which we study here. We find two non-abelian BI-groups of orders 20 and 42 which are not CI-groups. We also list all BI-groups of orders up to 30.

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