scholarly journals The Laplacian energy of conjugacy class graph of some dihedral groups

2017 ◽  
Vol 13 (2) ◽  
Author(s):  
Rabiha Birkia Mahmoud ◽  
Nor Haniza Sarmin ◽  
Ahmad Erfanian ◽  
Amira Fadina Ahmad Fadzil
Keyword(s):  
Conjugacy Class ◽  
Dihedral Group ◽  
Dihedral Groups ◽  
Laplacian Eigenvalues ◽  
Laplacian Matrices ◽  
Laplacian Energy ◽  
The Absolute ◽  
The Difference ◽  
Class Graph

Let G be a dihedral group and Gamma its conjugacy class graph. The Laplacian energy of the graph, LE(Gamma) 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 number of vertices. In this research, the Laplacian matrices of the conjugacy class graph of some dihedral groups and its eigenvalues are first computed. Then, the Laplacian energy of this graph is determined.

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 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 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.

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 DETOUR ENERGY OF COMPLEMENT OF SUBGROUP GRAPH OF DIHEDRAL GROUP

2017 ◽  
Vol 1 (2) ◽  
Author(s):  
Abdussakir Abdussakir
Keyword(s):  
Dihedral Group ◽  
Normal Subgroups ◽  
Dihedral Groups ◽  
The Absolute

Study on the energy of a graph becomes a topic of great interest. One is the detour energy which is the sum of the absolute values of all eigenvalue of the detour matrix of a graph. Graphs obtained from a group also became a study that attracted the attention of many researchers. This article discusses the subgroup graph for several normal subgroups of dihedral groups. The discussion focused on the detour energy of complement of subgroup graph of dihedral group

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 Wiener and Zagreb Indices of Conjugacy Class Graph of the Dihedral Groups

MATEMATIKA ◽  
2019 ◽  
Vol 35 (1) ◽  
pp. 51-57 ◽  
Author(s):  
Nur Idayu Alimon ◽  
Nor Haniza Sarmin ◽  
Ahmad Erfanian
Keyword(s):  
Conjugacy Class ◽  
Chemical Properties ◽  
Wiener Index ◽  
Topological Indices ◽  
Dihedral Groups ◽  
Zagreb Index ◽  
Vertex Set ◽  
Class Graph ◽  
The First Zagreb Index ◽  
Group Of Symmetries

Topological indices are numerical values that can be analysed to predict the chemical properties of the molecular structure and the topological indices are computed for a graph related to groups. Meanwhile, the conjugacy class graph of  is defined as a graph with a vertex set represented by the non-central conjugacy classes of . Two distinct vertices are connected if they have a common prime divisor. The main objective of this article is to find various topological indices including the Wiener index, the first Zagreb index and the second Zagreb index for the conjugacy class graph of dihedral groups of order  where the dihedral group is the group of symmetries of regular polygon, which includes rotations and reflections. Many topological indices have been determined for simple and connected graphs in general but not graphs related to groups.  In this article, the Wiener index and Zagreb index of conjugacy class graph of dihedral groups are generalized.

Dihedral Groups of Order 2p of Automorphisms of Compact Riemann Surfaces of Genus p – 1

2015 ◽  
Vol 58 (1) ◽  
pp. 196-206
Author(s):  
Qingjie Yang ◽  
Weiting Zhong
Keyword(s):  
Automorphism Group ◽  
Conjugacy Class ◽  
Riemann Surfaces ◽  
Symplectic Group ◽  
Dihedral Group ◽  
Dihedral Groups ◽  
Analytic Automorphism ◽  
Compact Riemann Surfaces

AbstractIn this paper we prove that there is only one conjugacy class of dihedral group of order 2p in the 2(p – 1) × 2(p – 1) integral symplectic group that can be realized by an analytic automorphism group of compact connected Riemann surfaces of genus p – 1. A pair of representative generators of the realizable class is also given.

Laplacian spectra of Coprime Graph of finite cyclic and Dihedral groups

2020 ◽  
pp. 2150020
Author(s):  
Subarsha Banerjee
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Characteristic Polynomial ◽  
Dihedral Group ◽  
Dihedral Groups ◽  
Vertex Connectivity ◽  
Laplacian Eigenvalues ◽  
Laplacian Spectra ◽  
A Finite Group ◽  
Second Largest Eigenvalue

Let [Formula: see text] denote the Coprime Graph of a finite group [Formula: see text]. In this paper we study the Laplacian eigenvalues of the Coprime Graph of the finite cyclic group [Formula: see text] and the Dihedral group [Formula: see text] where [Formula: see text]. We find the characteristic polynomial of [Formula: see text] for any [Formula: see text] and determine the eigenvalues of [Formula: see text] for [Formula: see text] where [Formula: see text] are primes and [Formula: see text] is a positive integer. We characterize the values of [Formula: see text] for which algebraic and vertex connectivity of [Formula: see text] are equal. We also discuss about the largest and the second largest eigenvalue of [Formula: see text]. Finally, the spectra of [Formula: see text] has been determined for [Formula: see text] where [Formula: see text] are primes and [Formula: see text] is a positive integer.

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