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.

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 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.

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.

scholarly journals Rainbow connectivity of the non-commuting graph of a finite group

2016 ◽  
Vol 15 (07) ◽  
pp. 1650127 ◽  
Author(s):  
Yulong Wei ◽  
Xuanlong Ma ◽  
Kaishun Wang
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Positive Integer ◽  
Abelian Groups ◽  
Connection Number ◽  
Commuting Graph ◽  
Rainbow Connection Number ◽  
Rainbow Connection ◽  
Vertex Set ◽  
A Finite Group

Let [Formula: see text] be a finite non-abelian group. The non-commuting graph [Formula: see text] of [Formula: see text] has the vertex set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if [Formula: see text], where [Formula: see text] is the center of [Formula: see text]. We prove that the rainbow [Formula: see text]-connectivity of [Formula: see text] is [Formula: see text]. In particular, the rainbow connection number of [Formula: see text] is [Formula: see text]. Moreover, for any positive integer [Formula: see text], we prove that there exist infinitely many non-abelian groups [Formula: see text] such that the rainbow [Formula: see text]-connectivity of [Formula: see text] is [Formula: see text].

Non-commuting graphs of certain almost simple groups

2019 ◽  
Vol 12 (05) ◽  
pp. 1950081
Author(s):  
M. Jahandideh ◽  
R. Modabernia ◽  
S. Shokrolahi
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Simple Group ◽  
Simple Groups ◽  
Almost Simple Group ◽  
Almost Simple Groups ◽  
Commuting Graph ◽  
Vertex Set

Let [Formula: see text] be a non-abelian finite group and [Formula: see text] be the center of [Formula: see text]. The non-commuting graph, [Formula: see text], associated to [Formula: see text] is the graph whose vertex set is [Formula: see text] and two distinct vertices [Formula: see text] are adjacent if and only if [Formula: see text]. We conjecture that if [Formula: see text] is an almost simple group and [Formula: see text] is a non-abelian finite group such that [Formula: see text], then [Formula: see text]. Among other results, we prove that if [Formula: see text] is a certain almost simple group and [Formula: see text] is a non-abelian group with isomorphic non-commuting graphs, then [Formula: see text].

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].

Characterization of Aut(J2) and Aut(McL) by Their Non-commuting Graphs

2011 ◽  
Vol 18 (02) ◽  
pp. 327-332 ◽  
Author(s):  
Lingli Wang ◽  
Wujie Shi
Keyword(s):  
Abelian Group ◽  
Commuting Graph ◽  
Vertex Set

For a non-abelian group G, we associate the non-commuting graph ∇ (G) whose vertex set is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. In this paper, we prove that Aut (J2) and Aut (McL) are characterized by their non-commuting graphs.

THE EXISTENCE OR NONEXISTENCE OF NON-COMMUTING GRAPHS WITH PARTICULAR PROPERTIES

2013 ◽  
Vol 13 (01) ◽  
pp. 1350064 ◽  
Author(s):  
M. AKBARI ◽  
A. R. MOGHADDAMFAR
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Strongly Regular Graph ◽  
Complete Multipartite Graph ◽  
Commuting Graph ◽  
Central Elements ◽  
Vertex Set ◽  
Strongly Regular ◽  
Multipartite Graph ◽  
Complete Multipartite

We consider the non-commuting graph ∇(G) of a non-abelian finite group G; its vertex set is G\Z(G), the set of non-central elements of G, and two distinct vertices x and y are joined by an edge if [x, y] ≠ 1. We determine the structure of any finite non-abelian group G (up to isomorphism) for which ∇(G) is a complete multipartite graph (see Propositions 3 and 4). It is also shown that a non-commuting graph is a strongly regular graph if and only if it is a complete multipartite graph. Finally, it is proved that there is no non-abelian group whose non-commuting graph is self-complementary and n-cube.

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>

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