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 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 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 UNITS IN F2D2p

2013 ◽  
Vol 13 (02) ◽  
pp. 1350090 ◽  
Author(s):  
KULDEEP KAUR ◽  
MANJU KHAN
Keyword(s):  
Finite Field ◽  
Group Algebra ◽  
Dihedral Group ◽  
Unit Group ◽  
Normal Subgroups ◽  
Canonical Involution ◽  
Unitary Subgroup ◽  
Bicyclic Units

Let p be an odd prime, D2p be the dihedral group of order 2p, and F2 be the finite field with two elements. If * denotes the canonical involution of the group algebra F2D2p, then bicyclic units are unitary units. In this note, we investigate the structure of the group [Formula: see text], generated by the bicyclic units of the group algebra F2D2p. Further, we obtain the structure of the unit group [Formula: see text] and the unitary subgroup [Formula: see text], and we prove that both [Formula: see text] and [Formula: see text] are normal subgroups of [Formula: see text].

scholarly journals Hurwitz Equivalence in Tuples of Generalized Quaternion Groups and Dihedral Groups

10.37236/804 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Xiang-dong Hou
Keyword(s):  
Dihedral Group ◽  
Quaternion Group ◽  
Dihedral Groups ◽  
Generalized Quaternion Group ◽  
Hurwitz Action ◽  
Generalized Quaternion

Let $Q_{2^m}$ be the generalized quaternion group of order $2^m$ and $D_N$ the dihedral group of order $2N$. We classify the orbits in $Q_{2^m}^n$ and $D_{p^m}^n$ ($p$ prime) under the Hurwitz action.

The power digraphs associated with generalized dihedral groups

2015 ◽  
Vol 07 (04) ◽  
pp. 1550057 ◽  
Author(s):  
Uzma Ahmad
Keyword(s):  
Dihedral Group ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Dihedral Groups ◽  
Degree Structure ◽  
Number Of Cycles ◽  
Necessary And Sufficient ◽  
Power Mapping

We study the digraphs based on dihedral group [Formula: see text] by using the power mapping, i.e., the set of vertices of these digraphs is [Formula: see text] and the set of edges is [Formula: see text]. These are called the power digraphs and denoted by [Formula: see text]. The cycle and in-degree structure of these digraphs are completely examined. This investigation leads to the derivation of various formulae regarding the number of cycle vertices, the length of the cycles, the number of cycles of certain lengths and the in-degrees of all vertices. We also establish necessary and sufficient conditions for a vertex to be a cycle vertex. The analysis of distance between vertices culminates at different expressions in terms of [Formula: see text] and [Formula: see text] to determine the heights of vertices, components and the power digraph itself. Moreover, all regular and semi-regular power digraphs [Formula: see text] are completely classified.

scholarly journals The Representations of Quantum Double of Dihedral Groups

2013 ◽  
Vol 20 (01) ◽  
pp. 95-108 ◽  
Author(s):  
Jingcheng Dong ◽  
Huixiang Chen
Keyword(s):  
Group Algebra ◽  
Dihedral Group ◽  
Closed Field ◽  
Quantum Double ◽  
Dihedral Groups ◽  
Characteristic P ◽  
Algebraically Closed Field ◽  
Finite Dimensional

Let k be an algebraically closed field of odd characteristic p, and let Dn be the dihedral group of order 2n such that p|2n. Let D(kDn) denote the quantum double of the group algebra kDn. In this paper, we describe the structures of all finite-dimensional indecomposable left D(kDn)-modules, equivalently, of all finite-dimensional indecomposable Yetter-Drinfeld kDn-modules, and classify them.

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 The Non-Normal Subgroup Graph of Some Dihedral Groups

2020 ◽  
pp. 1-7
Author(s):  
Nabilah Fikriah Rahin ◽  
Nor Haniza Sarmin ◽  
Sheila Ilangovan
Keyword(s):  
Normal Subgroup ◽  
Dihedral Group ◽  
Simple Graph ◽  
Dihedral Groups ◽  
Vertex Set

Let G be a finite groupand H is a subgroup of G. The subgroup graph of H in G is defined as a directed simple graph with vertex set G and two distinct elements x and y are adjacent if and only if xy∈H. In this paper, the work on subgroup graph is extended by defining a new graph called the non-normal subgroup graph. The subgroup graph is determined for some dihedral groupsof order 2n when the subgroup is non-normal. Keywords: subgroup graph; non-normal subgroup; dihedral group

scholarly journals Total characters and Chebyshev polynomials

2003 ◽  
Vol 2003 (38) ◽  
pp. 2447-2453 ◽  
Author(s):  
Eirini Poimenidou ◽  
Homer Wolfe
Keyword(s):  
Finite Group ◽  
Chebyshev Polynomials ◽  
Irreducible Character ◽  
Dihedral Group ◽  
Dihedral Groups ◽  
Irreducible Characters ◽  
Integer Coefficients ◽  
Complete Answer ◽  
A Finite Group

The total characterτof a finite groupGis defined as the sum of all the irreducible characters ofG. K. W. Johnson asks when it is possible to expressτas a polynomial with integer coefficients in a single irreducible character. In this paper, we give a complete answer to Johnson's question for all finite dihedral groups. In particular, we show that, when such a polynomial exists, it is unique and it is the sum of certain Chebyshev polynomials of the first kind in any faithful irreducible character of the dihedral groupG.

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