scholarly journals On Commuting Graphs for Elements of Order 3 in Symmetric Groups

10.37236/2362 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Athirah Nawawi ◽  
Peter Rowley
Keyword(s):  
Conjugacy Class ◽  
Symmetric Group ◽  
Symmetric Groups ◽  
Commuting Graph ◽  
Vertex Set

The commuting graph $\mathcal{C}(G,X)$, where $G$ is a group and $X$ is a subset of $G$, is the graph with vertex set $X$ and distinct vertices being joined by an edge whenever they commute. Here the diameter of $\mathcal{C}(G,X)$ is studied when $G$ is a symmetric group and $X$ a conjugacy class of elements of order $3$.

scholarly journals A Note on Commuting Graphs for Symmetric Groups

10.37236/95 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
C. Bates ◽  
D. Bundy ◽  
S. Hart ◽  
P. Rowley
Keyword(s):  
Conjugacy Class ◽  
Symmetric Group ◽  
Symmetric Groups ◽  
Commuting Graph ◽  
Vertex Set

The commuting graph ${\cal C}(G,X)$, where $G$ is a group and $X$ a subset of $G$, has $X$ as its vertex set with two distinct elements of $X$ joined by an edge when they commute in $G$. Here the diameter and disc structure of ${\cal C}(G,X)$ is investigated when $G$ is the symmetric group and $X$ a conjugacy class of $G$.

scholarly journals Commuting Graphs, C(G, X) in Symmetric Groups Sym(n) and Its Connectivity

Symmetry ◽  
2019 ◽  
Vol 11 (9) ◽  
pp. 1178
Author(s):  
Athirah Nawawi ◽  
Sharifah Kartini Said Husain ◽  
Muhammad Rezal Kamel Ariffin
Keyword(s):  
Conjugacy Class ◽  
Symmetric Group ◽  
Symmetric Groups ◽  
Commuting Graph ◽  
Commutativity Property

A commuting graph is a graph denoted by C ( G , X ) where G is any group and X, a subset of a group G, is a set of vertices for C ( G , X ) . Two distinct vertices, x , y ∈ X , will be connected by an edge if the commutativity property is satisfied or x y = y x . This study presents results for the connectivity of C ( G , X ) when G is a symmetric group of degree n, Sym ( n ) , and X is a conjugacy class of elements of order three in G.

scholarly journals The Topological Indices of the Non-commuting Graph for Symmetric Groups

2020 ◽  
pp. 1-5
Author(s):  
Nur Idayu Alimon ◽  
Nor Haniza Sarmin ◽  
Ahmad Erfanian
Keyword(s):  
Symmetric Group ◽  
Chemical Compound ◽  
Molecular Graph ◽  
Symmetric Groups ◽  
Wiener Index ◽  
Topological Indices ◽  
Zagreb Index ◽  
Commuting Graph ◽  
Central Elements ◽  
Vertex Set

Topological indices are the numerical values that can be calculated from a graph and it is calculated based on the molecular graph of a chemical compound. It is often used in chemistry to analyse the physical properties of the molecule which can be represented as a graph with a set of vertices and edges. Meanwhile, the non-commuting graph is the graph of vertex set whose vertices are non-central elements and two distinct vertices are joined by an edge if they do not commute. The symmetric group, denoted as S_n, is a set of all permutation under composition. In this paper, two of the topological indices, namely the Wiener index and the Zagreb index of the non-commuting graph for symmetric groups of order 6 and 24 are determined. Keywords: Wiener index; Zagreb index; non-commuting graph; symmetric groups

scholarly journals New characterization of symmetric groups of prime degree

2017 ◽  
Vol 9 (1) ◽  
pp. 5-12
Author(s):  
A. K. Asboei ◽  
R. Mohammadyari
Keyword(s):  
Conjugacy Class ◽  
Symmetric Group ◽  
Class Size ◽  
Symmetric Groups ◽  
Prime Degree ◽  
Conjugacy Class Size

Abstract In this paper, will show that a symmetric group of prime degree p ≥ 5 is recognizable by its order and a special conjugacy class size of (p − 1)!.

scholarly journals Commuting Involution Graphs for $3$-Dimensional Unitary Groups

10.37236/590 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Alistaire Everett
Keyword(s):  
Conjugacy Class ◽  
Unitary Group ◽  
Unitary Groups ◽  
Special Unitary Group ◽  
3 Dimensional ◽  
Commuting Graph ◽  
Vertex Set

For a group $G$ and $X$ a subset of $G$ the commuting graph of $G$ on $X$, denoted by $\cal{C}(G,X)$, is the graph whose vertex set is $X$ with $x,y\in X$ joined by an edge if $x\neq y$ and $x$ and $y$ commute. If the elements in $X$ are involutions, then $\cal{C}(G,X)$ is called a commuting involution graph. This paper studies $\cal{C}(G,X)$ when $G$ is a 3-dimensional projective special unitary group and $X$ a $G$-conjugacy class of involutions, determining the diameters and structure of the discs of these graphs.

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].

Conditions on the Edges and Vertices of Non-commuting Graph

2015 ◽  
Vol 74 (1) ◽  
Author(s):  
M. Jahandideh ◽  
M. R. Darafsheh ◽  
N. H. Sarmin ◽  
S. M. S. Omer
Keyword(s):  
Finite Group ◽  
Conjugacy Classes ◽  
Commuting Graph ◽  
Vertex Set ◽  
A Finite Group

Abstract - Let G􀡳 be a non- abelian finite group. The non-commuting graph ,􀪡is defined as a graph with a vertex set􀡳 − G-Z(G)􀢆in which two vertices x􀢞 and y􀢟 are joined if and only if xy􀢞􀢟 ≠ yx􀢟􀢞.  In this paper, we invest some results on the number of edges set , the degree of avertex of non-commuting graph and the number of conjugacy classes of a finite group. In order that if 􀪡􀡳non-commuting graph of H ≅ non - commuting graph of G􀪡􀡴,H 􀡴 is afinite group, then |G􀡳| = |H􀡴| .

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.

scholarly journals On diameters estimations of the commuting graphs of Sylow $p$-subgroups of the symmetric groups

2013 ◽  
Vol 5 (1) ◽  
pp. 70-78 ◽  
Author(s):  
Yu.Yu. Leshchenko ◽  
L.V. Zoria
Keyword(s):  
Undirected Graph ◽  
Symmetric Groups ◽  
Commuting Graph ◽  
Central Elements

The commuting graph of a group $G$ is an undirected graph whose vertices are non-central elements of $G$ and two distinct vertices $x,y$ are adjacent if and only if $xy=yx$. This article deals with the properties of the commuting graphs of Sylow $p$-subgroups of the symmetric groups. We define conditions of connectedness of respective graphs and give estimations of the diameters if graph is connected.

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