On the non-commuting graph in finite Moufang loops

2018 ◽  
Vol 17 (04) ◽  
pp. 1850070
Author(s):  
Karim Ahmadidelir
Keyword(s):  
Abelian Group ◽  
Nilpotent Groups ◽  
Moufang Loop ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Commutative Moufang Loop ◽  
Commuting Graph ◽  
Moufang Loops ◽  
Central Elements ◽  
Vertex Set

The non-commuting graph associated to a non-abelian group [Formula: see text], [Formula: see text], is a graph with vertex set [Formula: see text] where distinct non-central elements [Formula: see text] and [Formula: see text] of [Formula: see text] are joined by an edge if and only if [Formula: see text]. The non-commuting graph of a non-abelian finite group has received some attention in existing literature. Recently, many authors have studied the non-commuting graph associated to a non-abelian group. In particular, the authors put forward the following conjectures: Conjecture 1. Let [Formula: see text] and [Formula: see text] be two non-abelian finite groups such that [Formula: see text]. Then [Formula: see text]. Conjecture 2 (AAM’s Conjecture). Let [Formula: see text] be a finite non-abelian simple group and [Formula: see text] be a group such that [Formula: see text]. Then [Formula: see text]. Some authors have proved the first conjecture for some classes of groups (specially for all finite simple groups and non-abelian nilpotent groups with irregular isomorphic non-commuting graphs) but in [Moghaddamfar, About noncommuting graphs, Sib. Math. J. 47(5) (2006) 911–914], Moghaddamfar has shown that it is not true in general with some counterexamples to this conjecture. On the other hand, Solomon and Woldar proved the second conjecture, in [R. Solomon and A. Woldar, Simple groups are characterized by their non-commuting graph, J. Group Theory 16 (2013) 793–824]. In this paper, we will define the same concept for a finite non-commutative Moufang loop [Formula: see text] and try to characterize some finite non-commutative Moufang loops with their non-commuting graph. Particularly, we obtain examples of finite non-associative Moufang loops and finite associative Moufang loops (groups) of the same order which have isomorphic non-commuting graphs. Also, we will obtain some results related to the non-commuting graph of a finite non-commutative Moufang loop. Finally, we give a conjecture stating that the above result is true for all finite simple Moufang loops.

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].

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.

On the Clique Numbers of Non-commuting Graphs of Certain Groups

2010 ◽  
Vol 17 (04) ◽  
pp. 611-620 ◽  
Author(s):  
A. Abdollahi ◽  
A. Azad ◽  
A. Mohammadi Hassanabadi ◽  
M. Zarrin
Keyword(s):  
Solvable Groups ◽  
Maximum Size ◽  
Clique Number ◽  
Special Linear Group ◽  
Simple Graph ◽  
Simple Groups ◽  
Projective Special Linear Group ◽  
Commuting Graph ◽  
Central Elements ◽  
Vertex Set

Let G be a non-abelian group. The non-commuting graph [Formula: see text] of G is defined as the graph whose vertex set is the non-central elements of G and two vertices are joint if and only if they do not commute. In a finite simple graph Γ, the maximum size of complete subgraphs of Γ is called the clique number of Γ and denoted by ω(Γ). In this paper, we characterize all non-solvable groups G with [Formula: see text], where 57 is the clique number of the non-commuting graph of the projective special linear group PSL (2,7). We also determine [Formula: see text] for all finite minimal simple groups G.

scholarly journals The Non-Commuting, Non-Generating Graph of a Nilpotent Group

10.37236/9802 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Peter Cameron ◽  
Saul Freedman ◽  
Colva Roney-Dougal
Keyword(s):  
Maximal Subgroup ◽  
Nilpotent Group ◽  
Commuting Graph ◽  
Generating Graph ◽  
Central Elements ◽  
Vertex Set ◽  
Infinite Case ◽  
The Difference ◽  
The Relationship

For a nilpotent group $G$, let $\Xi(G)$ be the difference between the complement of the generating graph of $G$ and the commuting graph of $G$, with vertices corresponding to central elements of $G$ removed. That is, $\Xi(G)$ has vertex set $G \setminus Z(G)$, with two vertices adjacent if and only if they do not commute and do not generate $G$. Additionally, let $\Xi^+(G)$ be the subgraph of $\Xi(G)$ induced by its non-isolated vertices. We show that if $\Xi(G)$ has an edge, then $\Xi^+(G)$ is connected with diameter $2$ or $3$, with $\Xi(G) = \Xi^+(G)$ in the diameter $3$ case. In the infinite case, our results apply more generally, to any group with every maximal subgroup normal. When $G$ is finite, we explore the relationship between the structures of $G$ and $\Xi(G)$ in more detail.

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.

scholarly journals The independence polynomial of inverse commuting graph of dihedral groups

2020 ◽  
Vol 16 (1) ◽  
pp. 115-120
Author(s):  
Aliyu Suleiman ◽  
Aliyu Ibrahim Kiri
Keyword(s):  
Identity Element ◽  
Independent Set ◽  
Independent Sets ◽  
Dihedral Groups ◽  
Independence Polynomial ◽  
Independence Polynomials ◽  
Commuting Graph ◽  
Central Elements ◽  
Vertex Set

Set of vertices not joined by an edge in a graph is called the independent set of the graph. The independence polynomial of a graph is a polynomial whose coefficient is the number of independent sets in the graph. In this research, we introduce and investigate the inverse commuting graph of dihedral groups (D2N) denoted by GIC. It is a graph whose vertex set consists of the non-central elements of the group and for distinct  x,y, E D2N, x and y are adjacent if and only if xy = yx = 1  where 1 is the identity element. The independence polynomials of the inverse commuting graph for dihedral groups are also computed. A formula for obtaining such polynomials without getting the independent sets is also found, which was used to compute for dihedral groups of order 18 up to 32.

scholarly journals Radius, Diameter, Multiplisitas Sikel, dan Dimensi Metrik Graf Komuting dari Grup Dihedral

2017 ◽  
Vol 3 (1) ◽  
pp. 1-4
Author(s):  
Abdussakir Abdussakir
Keyword(s):  
Abelian Group ◽  
Dihedral Group ◽  
Metric Dimension ◽  
Commuting Graph ◽  
Vertex Set

Commuting graph C(G) of a non-Abelian group G is a graph that contains all elements of G as its vertex set and two distinct vertices in C(G) will be adjacent if they are commute in G. In this paper we discuss commuting graph of dihedral group D2n. We show radius, diameter, cycle multiplicity, and metric dimension of this commuting graph in several theorems with their proof.

A second grammar of associators

1978 ◽  
Vol 84 (3) ◽  
pp. 405-416 ◽  
Author(s):  
J. D. H. Smith
Keyword(s):  
Finite Number ◽  
Current Paper ◽  
Moufang Loop ◽  
Good Knowledge ◽  
Additional Material ◽  
Nilpotence Class ◽  
Number Of Generators ◽  
Commutative Moufang Loop ◽  
Moufang Loops

1. Introduction. This paper is concerned with proving identities in commutative Moufang loops. Many such identities were derived in chapter VIII of (1) in the course of demonstrating the local nilpotence of commutative Moufang loops. The results there are regarded as constituting the ‘first grammar of associators’: the reader is assumed to have a good knowledge of them. The current paper develops additional material required for the determination in (5) of the precise nilpotence class of the free commutative Moufang loop on any given finite number of generators. It is called a ‘grammar’ because it lists formal ways in which the language of associators works, and is merely meant to serve a reader of ‘literature’ in the language such as (5). However, it may be of interest for other purposes, such as answering Manin's question ((3), Vopros 10·3; (4), problem 10·2) on the 3-rank of the free commutative Moufang loop of exponent 3. There is also the problem raised below as to whether the Triple Argument Hypothesis is a consequence of the commutative Moufang loop laws. Finally, the Möbius function in Section 9 may tempt someone to look at lattice-theoretical aspects of associators.

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.

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