scholarly journals Common Neighborhood Energy of Commuting Graphs of Finite Groups

Symmetry ◽  
2021 ◽  
Vol 13 (9) ◽  
pp. 1651
Author(s):  
Rajat Kanti Nath ◽  
Walaa Nabil Taha Fasfous ◽  
Kinkar Chandra Das ◽  
Yilun Shang
Keyword(s):  
Abelian Group ◽  
Finite Groups ◽  
Regular Polygon ◽  
Undirected Graph ◽  
Abelian Groups ◽  
Simple Graph ◽  
Central Quotient ◽  
Vertex Set ◽  
The Common ◽  
Group Of Symmetries

The commuting graph of a finite non-abelian group G with center Z(G), denoted by Γc(G), is a simple undirected graph whose vertex set is G∖Z(G), and two distinct vertices x and y are adjacent if and only if xy=yx. Alwardi et al. (Bulletin, 2011, 36, 49-59) defined the common neighborhood matrix CN(G) and the common neighborhood energy Ecn(G) of a simple graph G. A graph G is called CN-hyperenergetic if Ecn(G)>Ecn(Kn), where n=|V(G)| and Kn denotes the complete graph on n vertices. Two graphs G and H with equal number of vertices are called CN-equienergetic if Ecn(G)=Ecn(H). In this paper we compute the common neighborhood energy of Γc(G) for several classes of finite non-abelian groups, including the class of groups such that the central quotient is isomorphic to group of symmetries of a regular polygon, and conclude that these graphs are not CN-hyperenergetic. We shall also obtain some pairs of finite non-abelian groups such that their commuting graphs are CN-equienergetic.

scholarly journals Common neighborhood spectrum of commuting graphs of finite groups

2021 ◽  
Vol 32 (1) ◽  
pp. 33-48
Author(s):  
W. N. T. Fasfous ◽  
◽  
R. Sharafdini ◽  
R. K. Nath ◽  
◽  
...  
Keyword(s):  
Abelian Group ◽  
Finite Groups ◽  
Undirected Graph ◽  
Abelian Groups ◽  
Commuting Graph ◽  
Vertex Set ◽  
The Common

The commuting graph of a finite non-abelian group G with center Z(G), denoted by Γc(G), is a simple undirected graph whose vertex set is G∖Z(G), and two distinct vertices x and y are adjacent if and only if xy=yx. In this paper, we compute the common neighborhood spectrum of commuting graphs of several classes of finite non-abelian groups and conclude that these graphs are CN-integral.

Connectivity and planarity of g-noncommuting graph of finite groups

2018 ◽  
Vol 17 (06) ◽  
pp. 1850107
Author(s):  
Mahboube Nasiri ◽  
Ahmad Erfanian ◽  
Abbas Mohammadian
Keyword(s):  
Abelian Group ◽  
Finite Groups ◽  
Undirected Graph ◽  
Abelian Groups ◽  
Nonidentity Element ◽  
Noncommuting Graph

Let [Formula: see text] be a finite non-abelian group and [Formula: see text] be its center. For a fixed nonidentity element [Formula: see text] of [Formula: see text], the [Formula: see text]-noncommuting graph of [Formula: see text], denoted by [Formula: see text], is a simple undirected graph in which its vertices are [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if [Formula: see text] and [Formula: see text]. In this paper, we discuss about connectivity of [Formula: see text] and determine all finite non-abelian groups such that their [Formula: see text]-noncommuting graphs are 1-planar, toroidal or projective.

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].

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 Regularity of extended conjugate graphs of finite groups

2022 ◽  
Vol 7 (4) ◽  
pp. 5480-5498
Author(s):  
Piyapat Dangpat ◽  
◽  
Teerapong Suksumran ◽  
Keyword(s):  
Finite Groups ◽  
Necessary Conditions ◽  
Undirected Graph ◽  
Algebraic Property ◽  
Sufficient Conditions ◽  
Strong Connection ◽  
Theoretic Property ◽  
Graph Theoretic ◽  
Vertex Set ◽  
A Finite Group

<abstract><p>The extended conjugate graph associated to a finite group $ G $ is defined as an undirected graph with vertex set $ G $ such that two distinct vertices joined by an edge if they are conjugate. In this article, we show that several properties of finite groups can be expressed in terms of properties of their extended conjugate graphs. In particular, we show that there is a strong connection between a graph-theoretic property, namely regularity, and an algebraic property, namely nilpotency. We then give some sufficient conditions and necessary conditions for the non-central part of an extended conjugate graph to be regular. Finally, we study extended conjugate graphs associated to groups of order $ pq $, $ p^3 $, and $ p^4 $, where $ p $ and $ q $ are distinct primes.</p></abstract>

scholarly journals Tree-Antimagicness of Disconnected Graphs

2015 ◽  
Vol 2015 ◽  
pp. 1-4 ◽  
Author(s):  
Martin Bača ◽  
Zuzana Kimáková ◽  
Andrea Semaničová-Feňovčíková ◽  
Muhammad Awais Umar
Keyword(s):  
Arithmetic Progression ◽  
Disjoint Union ◽  
Simple Graph ◽  
Initial Term ◽  
Vertex Set ◽  
The Common ◽  
Disconnected Graphs

A simple graphGadmits anH-covering if every edge inE(G)belongs to a subgraph ofGisomorphic toH. The graphGis said to be (a,d)-H-antimagic if there exists a bijection from the vertex setV(G)and the edge setE(G)onto the set of integers1, 2, …,VG+E(G)such that, for all subgraphsH′ofGisomorphic toH, the sum of labels of all vertices and edges belonging toH′constitute the arithmetic progression with the initial termaand the common differenced.Gis said to be a super (a,d)-H-antimagic if the smallest possible labels appear on the vertices. In this paper, we study super tree-antimagic total labelings of disjoint union of graphs.

scholarly journals Solving a fixed number of equations over finite groups

2021 ◽  
Vol 82 (1) ◽  
Author(s):  
Philipp Nuspl
Keyword(s):  
Abelian Group ◽  
Finite Groups ◽  
Abelian Groups ◽  
Single Equation ◽  
Fixed Number ◽  
Polynomial Equations ◽  
Systems Of Equations ◽  
Semidirect Products ◽  
Equations Over Groups ◽  
Systems Of Polynomial Equations

AbstractWe investigate the complexity of solving systems of polynomial equations over finite groups. In 1999 Goldmann and Russell showed $$\mathrm {NP}$$ NP -completeness of this problem for non-Abelian groups. We show that the problem can become tractable for some non-Abelian groups if we fix the number of equations. Recently, Földvári and Horváth showed that a single equation over groups which are semidirect products of a p-group with an Abelian group can be solved in polynomial time. We generalize this result and show that the same is true for systems with a fixed number of equations. This shows that for all groups for which the complexity of solving one equation has been proved to be in $$\mathrm {P}$$ P so far, solving a fixed number of equations is also in $$\mathrm {P}$$ P . Using the collecting procedure presented by Horváth and Szabó in 2006, we furthermore present a faster algorithm to solve systems of equations over groups of order pq.

scholarly journals Spectrum of commuting graphs of some classes of finite groups

MATEMATIKA ◽  
2017 ◽  
Vol 33 (1) ◽  
pp. 87 ◽  
Author(s):  
Rajat Kanti Nath ◽  
Jutirekha Dutta
Keyword(s):  
Abelian Group ◽  
Symmetric Group ◽  
Finite Groups ◽  
Abelian Groups ◽  
Commuting Graph

In this paper, we initiate the study of spectrum of the commuting graphs of finite non-abelian groups. We first compute the spectrum of this graph for several classes of finite groups, in particular AC-groups. We show that the commuting graphs of finite non-abelian AC-groups are integral. We also show that the commuting graph of a finite non-abelian group G is integral if G is not isomorphic to the symmetric group of degree 4 and the commuting graph of G is planar. Further, it is shown that the commuting graph of G is integral if its commuting graph is toroidal.

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].

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