THE CONJUGACY CLASSES OF SOME FINITE METABELIAN GROUPS AND THEIR RELATED GRAPHS

2016 ◽  
Vol 79 (1) ◽  
Author(s):  
Nor Haniza Sarmin ◽  
Ain Asyikin Ibrahim ◽  
Alia Husna Mohd Noor ◽  
Sanaa Mohamed Saleh Omer
Keyword(s):  
Graph Theory ◽  
Conjugacy Class ◽  
Dihedral Group ◽  
Clique Number ◽  
Conjugacy Classes ◽  
Quaternion Group ◽  
Metabelian Groups ◽  
Dominating Number ◽  
Graph Properties ◽  
Class Graph

In this paper, the conjugacy classes of three metabelian groups, namely the Quasi-dihedral group, Dihedral group and Quaternion group of order 16 are computed. The obtained results are then applied to graph theory, more precisely to conjugate graph and conjugacy class graph. Some graph properties such as chromatic number, clique number, dominating number and independent number are found.   

ON GRAPHS ASSOCIATED TO CONJUGACY CLASSES OF SOME THREE-GENERATOR GROUPS

2016 ◽  
Vol 79 (1) ◽  
Author(s):  
Nor Haniza Sarmin ◽  
Alia Husna Mohd Noor ◽  
Sanaa Mohamed Saleh Omer
Keyword(s):  
Conjugacy Class ◽  
Chromatic Number ◽  
Clique Number ◽  
Conjugacy Classes ◽  
Line Segments ◽  
Dominating Number ◽  
Commuting Graph ◽  
Independent Number ◽  
Graph Properties ◽  
Class Graph

A graph consists of points which are called vertices, and connections which are called edges, which are indicated by line segments or curves joining certain pairs of vertices.  In this paper, four types of graphs which are the commuting graph, non-commuting graph conjugate graph and the conjugacy class graph for some three-generator groups are discussed. Some of the graph properties are also found which include the independent number, chromatic number, clique number and dominating number.

scholarly journals The conjugacy class graphs of non-abelian 3-groups

2020 ◽  
Vol 16 (3) ◽  
pp. 297-299
Author(s):  
Athirah Zulkarnain ◽  
Nor Haniza Sarmin ◽  
Hazzirah Izzati Mat Hassim
Keyword(s):  
Conjugacy Class ◽  
Chromatic Number ◽  
Simple Graph ◽  
Conjugacy Classes ◽  
Prime Power Order ◽  
Complete Graphs ◽  
Dominating Number ◽  
Minimum Number ◽  
Vertex Set ◽  
Class Graph

A graph is formed by a pair of vertices and edges. It can be related to groups by using the groups’ properties for its vertices and edges. The set of vertices of the graph comprises the elements or sets from the group while the set of edges of the graph is the properties and condition for the graph. A conjugacy class of an element  is the set of elements that are conjugated with . Any element of a group , labelled as , is conjugated to  if it satisfies  for some elements  in  with its inverse . A conjugacy class graph of a group   is defined when its vertex set is the set of non-central conjugacy classes of  . Two distinct vertices   and   are connected by an edge if and only if their cardinalities are not co-prime, which means that the order of the conjugacy classes of  and  have common factors. Meanwhile, a simple graph is the graph that contains no loop and no multiple edges. A complete graph is a simple graph in which every pair of distinct vertices is adjacent. Moreover, a  -group is the group with prime power order. In this paper, the conjugacy class graphs for some non-abelian 3-groups are determined by using the group’s presentations and the definition of conjugacy class graph. There are two classifications of the non-abelian 3-groups which are used in this research. In addition, some properties of the conjugacy class graph such as the chromatic number, the dominating number, and the diameter are computed. A chromatic number is the minimum number of vertices that have the same colours where the adjacent vertices have distinct colours. Besides, a dominating number is the minimum number of vertices that is required to connect all the vertices while a diameter is the longest path between any two vertices. As a result of this research, the conjugacy class graphs of these groups are found to be complete graphs with chromatic number, dominating number and diameter that are equal to eight, one and one, respectively.

scholarly journals The independence and clique polynomial of the conjugacy class graph of dihedral group

2018 ◽  
Vol 14 ◽  
pp. 434-438
Author(s):  
Nabilah Najmuddin ◽  
Nor Haniza Sarmin ◽  
Ahmad Erfanian ◽  
Hamisan Rahmat
Keyword(s):  
Conjugacy Class ◽  
Complete Graph ◽  
Dihedral Group ◽  
Independent Set ◽  
Independent Sets ◽  
Conjugacy Classes ◽  
Dihedral Groups ◽  
Independence Polynomial ◽  
Independence Polynomials ◽  
Class Graph

The independence and clique polynomial are two types of graph polynomial that store combinatorial information of a graph. The independence polynomial of a graph is the polynomial in which its coefficients are the number of independent sets in the graph. The independent set of a graph is a set of vertices that are not adjacent. The clique polynomial of a graph is the polynomial in which its coefficients are the number of cliques in the graph. The clique of a graph is a set of vertices that are adjacent. Meanwhile, a graph of group G is called conjugacy class graph if the vertices are non-central conjugacy classes of G and two distinct vertices are connected if and only if their class cardinalities are not coprime. The independence and clique polynomial of the conjugacy class graph of a group G can be obtained by considering the polynomials of complete graph or polynomials of union of some graphs. In this research, the independence and clique polynomials of the conjugacy class graph of dihedral groups of order 2n are determined based on three cases namely when n is odd, when n and n/2 are even, and when n is even and n/2 is odd. For each case, the results of the independence polynomials are of degree two, one and two, and the results of the clique polynomials are of degree (n-1)/2, (n+2)/2 and (n-2)/2, respectively.

scholarly journals Related graphs of the conjugacy classes of a 3-generator 5-group

2018 ◽  
Vol 14 ◽  
pp. 454-456
Author(s):  
Alia Husna Mohd Noor ◽  
Nor Haniza Sarmin ◽  
Hamisan Rahmat
Keyword(s):  
Graph Theory ◽  
Conjugacy Class ◽  
Group Action ◽  
Equivalence Relation ◽  
Complete Graph ◽  
Undirected Graph ◽  
Conjugacy Classes ◽  
Group Presentation ◽  
Class Graph

The study on conjugacy class has started since 1968. A conjugacy class is defined as an equivalence class under the equivalence relation of being conjugate. In this research, let be a 3-generator 5-group and the scope of the graphs is a simple undirected graph. This paper focuses on the determination of the conjugacy classes of where the set omega is the subset of all commuting elements in the group. The elements of the group with order 5 are identified from the group presentation. The pair of elements are formed in the form of  which is of size two where  and  commute. In addition, the results on conjugacy classes of are applied into graph theory. The determination of the set omega is important in the computation of conjugacy classes in order to find the generalized conjugacy class graph and orbit graph. The group action that is considered to compute the conjugacy classes is conjugation action. Based on the computation, the generalized conjugacy class graph and orbit graph turned out to be a complete graph.

scholarly journals On Some Graphs of Finite Metabelian Groups of Order Less Than 24

MATEMATIKA ◽  
2019 ◽  
Vol 35 (2) ◽  
pp. 237-247
Author(s):  
Ibrahim Gambo ◽  
Nor Haniza Sarmin ◽  
Sanaa Mohamed Saleh Omer
Keyword(s):  
Normal Subgroup ◽  
Conjugacy Class ◽  
Finite Groups ◽  
Metabelian Group ◽  
Factor Group ◽  
Conjugacy Classes ◽  
Metabelian Groups ◽  
Abelian Normal Subgroup ◽  
Class Graph

In this work, a non-abelian metabelian group is represented by G while represents conjugacy class graph. Conjugacy class graph of a group is that graph associated with the conjugacy classes of the group. Its vertices are the non-central conjugacy classes of the group, and two distinct vertices are joined by an edge if their cardinalities are not coprime. A group is referred to as metabelian if there exits an abelian normal subgroup in which the factor group is also abelian. It has been proven earlier that 25 non-abelian metabelian groups which have order less than 24, which are considered in this work, exist. In this article, the conjugacy class graphs of non-abelian metabelian groups of order less than 24 are determined as well as examples of some finite groups associated to other graphs are given.

scholarly journals The Laplacian Energy of Conjugacy Class Graph of Some Finite Groups

MATEMATIKA ◽  
2019 ◽  
Vol 35 (1) ◽  
pp. 59-65
Author(s):  
Rabiha Mahmoud ◽  
Amira Fadina Ahmad Fadzil ◽  
Nor Haniza Sarmin ◽  
Ahmad Erfanian
Keyword(s):  
Conjugacy Class ◽  
Finite Groups ◽  
Dihedral Group ◽  
Dihedral Groups ◽  
Laplacian Eigenvalues ◽  
Laplacian Matrices ◽  
Laplacian Energy ◽  
The Difference ◽  
Generalized Quaternion ◽  
Class Graph

Let G be a dihedral group and its conjugacy class graph. The Laplacian energy of the graph, is defined as the sum of the absolute values of the difference between the Laplacian eigenvalues and the ratio of twice the edges number divided by the vertices number. In this research, the Laplacian matrices of the conjugacy class graph of some dihedral groups, generalized quaternion groups, quasidihedral groups and their eigenvalues are first computed. Then, the Laplacian energy of the graphs are determined.

scholarly journals The Energy of Conjugacy Classes Graphs of Some Order of Alternating Groups

2020 ◽  
Vol 7 (4) ◽  
pp. 62-71
Author(s):  
Zuzan Naaman Hassan ◽  
Nihad Titan Sarhan
Keyword(s):  
Conjugacy Class ◽  
Adjacency Matrix ◽  
Finite Graph ◽  
Conjugacy Classes ◽  
Alternating Group ◽  
Alternating Groups ◽  
Eigen Values ◽  
Finite Set ◽  
Class Graph ◽  
Square Matrix

The energy of a graph , is the sum of all absolute values of the eigen values of the adjacency matrix which is indicated by . An adjacency matrix is a square matrix used to represent of finite graph where the rows and columns consist of 0 or 1-entry depending on the adjacency of the vertices of the graph. The group of even permutations of a finite set is known as an alternating group  . The conjugacy class graph is a graph whose vertices are non-central conjugacy classes of a group , where two vertices are connected if their cardinalities are not coprime. In this paper, the conjugacy class of alternating group  of some order for   and their energy are computed. The Maple2019 software and Groups, Algorithms, and Programming (GAP) are assisted for computations.

scholarly journals On the generalized conjugacy class graph of some dihedral groups

2017 ◽  
Vol 13 (2) ◽  
Author(s):  
Nurhidayah Zaid ◽  
Nor Haniza Sarmin ◽  
Hamisan Rahmat
Keyword(s):  
Conjugacy Class ◽  
Mathematical Structure ◽  
Dihedral Groups ◽  
Common Multiple ◽  
Graph Properties ◽  
Class Graph

A graph is a mathematical structure which consists of vertices and edges that is used to model relations between object. In this research, the generalized conjugacy class graph is constructed for some dihedral groups to show the relation between orbits and their cardinalities. In order to construct the graph, the probability that an element of the dihedral groups fixes a set must first be obtained. The set under this study is the set of all pairs of commuting elements in the form of (a,b) where a and b are elements of the dihedral groups and the lowest common multiple of the order of the elements is two. The orbits of the set are then computed using conjugation action. Based on the results obtained, the generalized conjugacy class graph is constructed and some graph properties are also found. 

scholarly journals The conjugacy class graph of some finite groups and its energy

2017 ◽  
Vol 13 (4) ◽  
pp. 659-665 ◽  
Author(s):  
Rabiha Mahmoud ◽  
Nor Haniza Sarmin ◽  
Ahmad Erfanian
Keyword(s):  
Conjugacy Class ◽  
General Formula ◽  
Finite Groups ◽  
Adjacency Matrix ◽  
Dihedral Group ◽  
Dihedral Groups ◽  
The Absolute ◽  
Generalized Quaternion ◽  
Class Graph

The energy of a graph which is denoted by  is defined to be the sum of the absolute values of the eigenvalues of its adjacency matrix. In this paper we present the concepts of conjugacy class graph of dihedral groups and introduce the general formula for the energy of the conjugacy class graph of dihedral groups. The energy of any dihedral group of order   in different cases, depends on the parity of   is proved in this paper. Also we introduce the general formula for the conjugacy class graph of generalized quaternion groups and quasidihedral groups.

scholarly journals General Form of Domination Polynomial for Two Types of Graphs Associated to Dihedral Groups

MATEMATIKA ◽  
2019 ◽  
Vol 35 (2) ◽  
pp. 149-155
Author(s):  
Nabilah Najmuddin ◽  
Nor Haniza Sarmin ◽  
Ahmad Erfanian
Keyword(s):  
Conjugacy Class ◽  
Finite Groups ◽  
Conjugacy Classes ◽  
Complete Graphs ◽  
Dominating Sets ◽  
Dihedral Groups ◽  
Graph Polynomial ◽  
Central Elements ◽  
Domination Polynomial ◽  
Class Graph

A domination polynomial is a type of graph polynomial in which its coefficients represent the number of dominating sets in the graph. There are many researches being done on the domination polynomial of some common types of graphs but not yet for graphs associated to finite groups. Two types of graphs associated to finite groups are the conjugate graph and the conjugacy class graph. A graph of a group G is called a conjugate graph if the vertices are non-central elements of G and two distinct vertices are adjacent if they are conjugate to each other. Meanwhile, a conjugacy class graph of a group G is a graph in which its vertices are the non-central conjugacy classes of G and two distinct vertices are connected if and only if their class cardinalities are not coprime. The conjugate and conjugacy class graph of dihedral groups can be expressed generally as a union of complete graphs on some vertices. In this paper, the domination polynomials are computed for the conjugate and conjugacy class graphs of the dihedral groups.

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