Dihedral Group Research Articles

A group theory based topology optimization scheme for the design of inhomogeneous waveguides with dihedral group symmetries

In this paper, we introduce a novel topology optimization scheme dedicated to designing waveguides with inhomogeneities holding rotation and reflection symmetries. To fully exploit the symmetry features of the guide, we build the scheme by first developing a new computational algorithm that combines the Finite Element Method (FEM) with the group representation theory (GRT), i.e., the GRT – FEM algorithm. This combination allows the decomposition of the eigenvalue problem in a classic FEM to orthogonal (decoupled) subproblems. The subproblems are of much smaller sizes and can be treated in parallel, which therefore leads to an improved solver efficiency by an order of magnitude. For a targeted mode profile, the involvement of the GRT does not only enable a pre-selection of the subproblems that are to be optimized, which further improves the computational speed, but also completely avoids potential mode degeneracies, which adversely affects the convergence of the optimization. To validate, the GRT – FEM algorithm is first tested against the results from a full FEM simulation for a guide with inhomogeneities with six-fold rotation and reflection symmetries. Then, as an illustration, the entire scheme is applied to achieve a desired eigenmode with a targeted propagation constant. The proposed scheme does not only provide an efficient design tool for exploiting interesting wave phenomena in waveguiding structures possessing rotation and reflection symmetries, but also lays down a theoretical foundation for systematically integrating symmetries with classic Computational Electromagnetics (CEM) algorithms.

Seidel Laplacian and Seidel Signless Laplacian Energies of Commuting Graph for Dihedral Groups

In this paper, we discuss the energy of the commuting graph. The vertex set of the graph is dihedral groups and the edges between two distinct vertices represent the commutativity of the group elements. The spectrum of the graph is associated with the Seidel Laplacian and Seidel signless Laplacian matrices. The results are similar to the well-known fact that the energies are not odd integers. We also highlight the relation that the Seidel signless Laplacian energy is never less than Seidel Laplacian energy. Ultimately, we classify the graphs according to the energy value as the hyperenergetic.

Nuclear Dimension of Subhomogeneous Twisted Groupoid C*-algebras and Dynamic Asymptotic Dimension

Abstract We characterize subhomogeneity for twisted étale groupoid $\text{C}^{*}$-algebras and obtain an upper bound on their nuclear dimension. As an application, we remove the principality assumption in recent results on upper bounds on the nuclear dimension of a twisted étale groupoid $\text{C}^{*}$-algebra in terms of the dynamic asymptotic dimension of the groupoid and the covering dimension of its unit space. As a non-principal example, we show that the dynamic asymptotic dimension of any minimal (not necessarily free) action of the infinite dihedral group $D_{\infty }$ on an infinite compact Hausdorff space $X$ is always one. So if we further assume that $X$ is second-countable and has finite covering dimension, then $C(X)\rtimes _{r} D_{\infty }$ has finite nuclear dimension and is classifiable by its Elliott invariant.

Dihedral Artin representations and CM fields

For a fixed CM field K K with maximal totally real subfield F F , we consider isomorphism classes of dihedral Artin representations of F F which are induced from K K , distinguishing between those which are “canonically” induced from K K and those which are “noncanonically” induced from K K . The latter can arise only for Artin representations with image isomorphic to the dihedral group of order 8. We show that asymptotically, the number of noncanonically induced isomorphism classes is always comparable to and in some cases exceeds the number of canonically induced ones.

Cayley regression problem of some groups

In this paper, we consider the Cayley regression problem of the cyclic groups and the dihedral groups. We give the sufficient and necessary conditions for the cyclic group ℤ n to be a Cayley regression. Moreover, the dihedral group D 2n is a 3-Cayley regression if n is a prime power.

THE NON-BRAID GRAPH OF DIHEDRAL GROUP Dn

THE NON-BRAID GRAPH OF DIHEDRAL GROUP Dn

Quark and lepton modular models from the binary dihedral flavor symmetry

Inspired by the structure of top-down derived models endowed with modular flavor symmetries, we investigate the yet phenomenologically unexplored binary dihedral group 2D3. After building the vector-valued modular forms in the representations of 2D3 with small modular weights, we systematically classify all (Dirac and Majorana) mass textures of fermions with fractional modular weights and all possible 2 + 1-family structures. This allows us to explore the parameter space of fermion models based on 2D3, aiming at a description of both quarks and leptons with a minimal number of parameters and best compatibility with observed data. We consider the separate possibilities of neutrino masses generated by either a type-I seesaw mechanism or the Weinberg operator. We identify a model that, besides fitting all known flavor observables, delivers predictions for six not-yet measured parameters and favors normal-ordered neutrino masses generated by the Weinberg operator. It would be interesting to figure out whether it is possible to embed our model within a top-down scheme, such as \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb{T}}^{2}/{\\mathbb{Z}}_{4}$$\\end{document} heterotic orbifold compactifications.

Weyl symmetry for curve counting invariants via spherical twists

We study the curve counting invariants of Calabi–Yau 3-folds via the Weyl reflection along a ruled divisor. We obtain a new rationality result and functional equation for the generating functions of Pandharipande–Thomas invariants. When the divisor arises as resolution of a curve of A 1 A_1 -singularities, our results match the rationality of the associated Calabi–Yau orbifold. The symmetry on generating functions descends from the action of an infinite dihedral group of derived auto-equivalences, which is generated by the derived dual and a spherical twist. Our techniques involve wall-crossing formulas and generalized Donaldson–Thomas invariants for surface-like objects.

Dihedral groups with the m-DCI property

Dihedral groups with the m-DCI property

Discrete-time quantum walks on Cayley graphs of Dihedral groups using generalized Grover coins

Discrete-time quantum walks on Cayley graphs of Dihedral groups using generalized Grover coins

Characteristic polynomial of maximum and minimum matrix of square power graph of dihedral group of order 2n with odd natural number n

Characteristic polynomial of maximum and minimum matrix of square power graph of dihedral group of order 2n with odd natural number n

Laplacian polynomial of square power graph of dihedral group of order 2n with even natural number n

Laplacian polynomial of square power graph of dihedral group of order 2n with even natural number n

Spectral Properties of Power Graph of Dihedral Groups

This paper focuses on the power graph of dihedral groups of order $2n$, $D_{2n}$, where $n \geq 3$. We show the characteristic polynomial of the power graph corresponding to the adjacency, Laplacian, signless Laplacian, and normalized form of these matrices.

The enumeration of equivalent classes of minimal general dihedral group codes

A group code is a linear code which can be realized as a two-sided ideal of a group algebra over a finite field. When the characteristic of the field is prime to the order of the group, we will give explicit expressions for primitive central idempotents in the group algebra, which enables us to determine the number of equivalent classes of minimal group codes. Then, we apply our formula to calculate the number of equivalent classes of minimal general dihedral group codes.

Counting and Equidistribution of Reciprocal Geodesics and Dihedral Groups

Abstract We study the growth of the number of conjugacy classes of infinite dihedral subgroups of lattices in $\operatorname{PSL}_{2}{\mathbb{R}}$, generalizing the earlier work of Sarnak [ 9] and Bourgain–Kontorovich [ 4] on the growth of the number of reciprocal geodesics on the modular surface. We also prove that reciprocal geodesics are equidistributed in the unit tangent bundle.

Characteristic Polynomial of Power Graph for Dihedral Groups Using Degree-Based Matrices

A fundamental feature of spectral graph theory is the correspondence between matrix and graph. As a result of this relation, the characteristic polynomial of the graph can be formulated. This research focuses on the power graph of dihedral groups using degree-based matrices. Throughout this paper, we formulate the characteristic polynomial of the power graph of dihedral groups based on seven types of graph matrices which include the maximum degree, the minimum degree, the greatest common divisor degree, the first Zagreb, the second Zagreb, the misbalance degree, and the Nirmala matrices.

Analysıs on Propertıes and Structure of Dıhedral Groups

The structure of groups plays an important role in the study of the nature of the groups. We examine some basic properties and structural characteristics of the dihedral group of degree n, where n is a natural number, by group-theoretic approach. We begin the exploration by providing a foundational understanding of dihedral groups, elucidating their definitions and essential properties. Furthermore, we investigated the algebraic and geometric aspects of these groups, highlighting their role in describing symmetries of n-gons and other mathematical entities. Special attention is given to the distinctive features that differentiate dihedral groups from other algebraic structures. The analytic expressions for the order of subgroups are obtained and the commutativity investigated. The groups are all represented for further analysis and applications.

The Topological Indices of the p-Subgroup Graph of Dihedral Groups

A topological index is a number generated from a molecular structure that signifies the molecule’s fundamental structural characteristics. This correlation between the index and various physical attributes is based on an algebraic quantity related to the chemical structure. Many topological indices, such as the Wiener index, first and second Zagreb indices, can be employed to determine various properties, including chemical activity, thermodynamic properties, physicochemical activity, and biological activity. Meanwhile, the p-subgroup graph of a group G is defined as a graph whose vertices represent the elements of the group, and two vertices are adjacent if and only if the order of the subgroup is a prime power. The main objective of this paper is to establish the general formula for certain topological indices, specifically the Wiener index, first Zagreb index, and second Zagreb index for the -subgroup graph associated with dihedral groups.

Sombor Index and Sombor Polynomial of the Noncommuting Graph Associated to Some Finite Groups

Sombor index is a newly developed degree-based topological index which involves the degree of the vertex in a simple connected graph. The Sombor index is known as the square root of the sum of the squared degrees of two adjacent vertices in a graph. Meanwhile, the noncommuting graph associated to a group is a graph where its vertices are the non-central elements of the group and two vertices are adjacent if and only if they do not commute. In this study, a new notion called the Sombor polynomial is introduced. Then, the general formula of the Sombor index and the Sombor polynomial of the noncommuting graph associated to some finite groups are determined by using their definitions and some preliminaries. The groups involved in this research are the dihedral groups, the quasidihedral groups, and the generalized quaternion groups. The results found can help the chemists and biologists to predict the chemical and physical properties of the molecules without involving any laboratory work.

On the Laplacian Energy of an Orbit Graph of Finite Groups

Let β H denote the orbit graph of a finite group H . Let ζ be the set of commuting elements in H with order two. An orbit graph is a simple undirected graph where non-central orbits are represented as vertices in ζ , and two vertices in ζ are connected by an edge if they are conjugate. In this article, we explore the Laplacian energy and signless Laplacian energy of orbit graphs associated with dihedral groups of order $2w$ and quaternion groups of order 2 w .