Exhaustion 2-subsets in dihedral groups of order 2p

2018 ◽  
Vol 11 (03) ◽  
pp. 1850047
Author(s):  
Denis Chee Keong Wong ◽  
Kuan Wai Wong ◽  
Wun-She Yap
Keyword(s):  
Positive Integer ◽  
Dihedral Group ◽  
Nonempty Subset ◽  
Arbitrary Subset ◽  
Dihedral Groups ◽  
Explicit Formulas ◽  
General Finding

Let [Formula: see text] be the dihedral group of order [Formula: see text], where [Formula: see text] is an odd prime. A nonempty subset [Formula: see text] of [Formula: see text] is said to be exhaustive if there exists a positive integer [Formula: see text] such that [Formula: see text] covers all elements in [Formula: see text]. The smallest such [Formula: see text] is called the exhaustion number of [Formula: see text]. In general, finding [Formula: see text] for an arbitrary subset [Formula: see text] of [Formula: see text] is an interesting but difficult task. In this paper, we classify all possible exhaustion 2-subsets in [Formula: see text] by considering a 2-subset [Formula: see text] of [Formula: see text] with either [Formula: see text] or [Formula: see text] and [Formula: see text]. Some explicit formulas for [Formula: see text] are identified and hence some bounds are derived to prove the existence for certain families of exhaustive sets.

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.

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 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 Cusp forms of weight one, quartic reciprocity and elliptic curves

1985 ◽  
Vol 98 ◽  
pp. 117-137 ◽  
Author(s):  
Noburo Ishii
Keyword(s):  
Positive Integer ◽  
Elliptic Curves ◽  
Galois Group ◽  
Cusp Form ◽  
Rational Number ◽  
Galois Extension ◽  
Dihedral Group ◽  
Cusp Forms ◽  
Two Dimensional ◽  
Weight One

Let m be a non-square positive integer. Let K be the Galois extension over the rational number field Q generated by and . Then its Galois group over Q is the dihedral group D4 of order 8 and has the unique two-dimensional irreducible complex representation ψ. In view of the theory of Hecke-Weil-Langlands, we know that ψ defines a cusp form of weight one (cf. Serre [6]).

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.

COMPUTING MANY MAXIMAL INDEPENDENT SETS FOR HYPERGRAPHS IN PARALLEL

2007 ◽  
Vol 17 (02) ◽  
pp. 141-152 ◽  
Author(s):  
LEONID KHACHIYAN ◽  
ENDRE BOROS ◽  
VLADIMIR GURVICH ◽  
KHALED ELBASSIONI
Keyword(s):  
Positive Integer ◽  
Nonempty Subset ◽  
Independent Sets ◽  
Deterministic Algorithm ◽  
Average Degree ◽  
Maximal Independent Sets

A hypergraph [Formula: see text] is called uniformly δ-sparse if for every nonempty subset X ⊆ V of vertices, the average degree of the sub-hypergraph of [Formula: see text] induced by X is at most δ. We show that there is a deterministic algorithm that, given a uniformly δ-sparse hypergraph [Formula: see text], and a positive integer k, outputs k or all minimal transversals for [Formula: see text] in O(δ log (1 + k) polylog (δ|V|))-time using |V|O( log δ)kO(δ) processors. Equivalently, the algorithm can be used to compute in parallel k or all maximal independent sets for [Formula: see text].

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 Subconvexity for a double Dirichlet series and non-vanishing of L-functions

2018 ◽  
Vol 14 (06) ◽  
pp. 1573-1604
Author(s):  
Alexander Dahl
Keyword(s):  
Functional Equation ◽  
Positive Integer ◽  
Dirichlet Series ◽  
Upper Bound ◽  
Dihedral Group ◽  
Central Point ◽  
Double Dirichlet Series ◽  
Dirichlet Characters ◽  
Equation Group

We study a double Dirichlet series of the form [Formula: see text], where [Formula: see text] and [Formula: see text] are quadratic Dirichlet characters with prime conductors [Formula: see text] and [Formula: see text] respectively. A functional equation group isomorphic to the dihedral group of order 6 continues the function meromorphically to [Formula: see text]. The developed theory is used to prove an upper bound for the smallest positive integer [Formula: see text] such that [Formula: see text] does not vanish. Additionally, a convexity bound at the central point is established to be [Formula: see text] and a subconvexity bound of [Formula: see text] is proven. An application of bounds at the central point to the non-vanishing problem is also discussed.

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.

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