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.

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.   

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 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 Graphs Related to Comaximal Ideals of a Commutative Ring

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Tongsuo Wu ◽  
Meng Ye ◽  
Dancheng Lu ◽  
Houyi Yu
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Clique Number ◽  
Induced Subgraph ◽  
The Core ◽  
Maximal Ideals ◽  
Graph Properties ◽  
Vertex Set

We study the co maximal graph Ω(R), the induced subgraph Γ(R) of Ω(R) whose vertex set is R∖(U(R)∪J(R)), and a retract Γr(R) of Γ(R), where R is a commutative ring. For a graph Γ(R) which contains a cycle, we show that the core of Γ(R) is a union of triangles and rectangles, while a vertex in Γ(R) is either an end vertex or a vertex in the core. For a nonlocal ring R, we prove that both the chromatic number and clique number of Γ(R) are identical with the number of maximal ideals of R. A graph Γr(R) is also introduced on the vertex set {Rx∣x∈R∖(U(R)∪J(R))}, and graph properties of Γr(R) are studied.

scholarly journals Recognition and Optimization Algorithms for P5-Free Graphs

Symmetry ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 304
Author(s):  
Mihai Talmaciu ◽  
Luminiţa Dumitriu ◽  
Ioan Şuşnea ◽  
Victor Lepin ◽  
László Barna Iantovics
Keyword(s):  
Chromatic Number ◽  
Dominating Set ◽  
Independent Set ◽  
Clique Number ◽  
Recognition Algorithm ◽  
Feedback Vertex Set ◽  
Free Graph ◽  
Dominating Number ◽  
Vertex Set ◽  
Np Complete

The weighted independent set problem on P 5 -free graphs has numerous applications, including data mining and dispatching in railways. The recognition of P 5 -free graphs is executed in polynomial time. Many problems, such as chromatic number and dominating set, are NP-hard in the class of P 5 -free graphs. The size of a minimum independent feedback vertex set that belongs to a P 5 -free graph with n vertices can be computed in O ( n 16 ) time. The unweighted problems, clique and clique cover, are NP-complete and the independent set is polynomial. In this work, the P 5 -free graphs using the weak decomposition are characterized, as is the dominating clique, and they are given an O ( n ( n + m ) ) recognition algorithm. Additionally, we calculate directly the clique number and the chromatic number; determine in O ( n ) time, the size of a minimum independent feedback vertex set; and determine in O ( n + m ) time the number of stability, the dominating number and the minimum clique cover.

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 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.

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 NORMAL SUBGROUPS WHOSE CONJUGACY CLASS GRAPH HAS DIAMETER THREE

2016 ◽  
Vol 94 (2) ◽  
pp. 266-272
Author(s):  
ANTONIO BELTRÁN ◽  
MARÍA JOSÉ FELIPE ◽  
CARMEN MELCHOR
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Conjugacy Class ◽  
Conjugacy Classes ◽  
Normal Subgroups ◽  
A Finite Group ◽  
Class Graph

Let $G$ be a finite group and let $N$ be a normal subgroup of $G$. We determine the structure of $N$ when the diameter of the graph associated to the $G$-conjugacy classes contained in $N$ is as large as possible, that is, equal to three.

scholarly journals Investigation of Commuting Graphs for Elements of Order 3 in Certain Leech Lattice Groups

2021 ◽  
pp. 2640-2652
Author(s):  
Duha Abbas Azeez ◽  
Ali Abd Aubad
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Clique Number ◽  
Leech Lattice ◽  
Commuting Graph ◽  
A Finite Group ◽  
Janko Group

      Assume that G is a finite group and X is a subset of G. The commuting graph is denoted by С(G,X) and has a set of vertices X with two distinct vertices x, y Î X, being connected together on the condition of xy = yx. In this paper, we investigate the structure of Ϲ(G,X) when G is a particular type of Leech lattice groups, namely Higman–Sims group HS and Janko group J2, along with  X as a G-conjugacy class of elements of order 3. We will pay particular attention to analyze the discs’ structure and determinate the diameters, girths, and clique number for these graphs.

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