Hermitian Adjacency Spectrum of Cayley Digraphs over Dihedral Group

2020 ◽  
Vol 27 (01) ◽  
pp. 121-130
Author(s):  
Honghai Li ◽  
Teng Yu
Keyword(s):  
Adjacency Matrix ◽  
Dihedral Group ◽  
Cayley Digraph ◽  
Adjacency Spectrum ◽  
Group D

We first study the spectrum of Hermitian adjacency matrix (H-spectrum) of Cayley digraphs X(D2n, S) on dihedral group D2n with |S| = 3. Then we show that all Cayley digraphs X(D2p, S) with |S| = 3 and p odd prime are Cay-DS, namely, for any Cayley digraph X(D2p, T), X(D2p, T) and X(D2p, S) having the same H-spectrum implies that they are isomorphic.

scholarly journals Groups all of whose undirected Cayley graphs are determined by their spectra

2016 ◽  
Vol 15 (09) ◽  
pp. 1650175 ◽  
Author(s):  
Alireza Abdollahi ◽  
Shahrooz Janbaz ◽  
Mojtaba Jazaeri
Keyword(s):  
Adjacency Matrix ◽  
Sylow Subgroup ◽  
Dihedral Group ◽  
Cayley Graphs ◽  
Solvable Groups ◽  
Adjacency Spectrum

The adjacency spectrum [Formula: see text] of a graph [Formula: see text] is the multiset of eigenvalues of its adjacency matrix. Two graphs with the same spectrum are called cospectral. A graph [Formula: see text] is “determined by its spectrum” (DS for short) if every graph cospectral to it is in fact isomorphic to it. A group is DS if all of its Cayley graphs are DS. A group [Formula: see text] is Cay-DS if every two cospectral Cayley graphs of [Formula: see text] are isomorphic. In this paper, we study finite DS groups and finite Cay-DS groups. In particular we prove that a finite DS group is solvable, and every non-cyclic Sylow subgroup of a finite DS group is of order [Formula: see text], [Formula: see text], [Formula: see text] or [Formula: see text]. We also give several infinite families of non-Cay-DS solvable groups. In particular we prove that there exist two cospectral non-isomorphic [Formula: see text]-regular Cayley graphs on the dihedral group of order [Formula: see text] for any prime [Formula: see text].

scholarly journals Some characterizatsion of coprime graph of dihedral group D 2n

2021 ◽  
Vol 1722 ◽  
pp. 012051
Author(s):  
A G Syarifudin ◽  
Nurhabibah ◽  
D P Malik ◽  
I G A W Wardhana
Keyword(s):  
Dihedral Group ◽  
Group D

scholarly journals Strong duality for metacyclic groups

2002 ◽  
Vol 73 (3) ◽  
pp. 377-392 ◽  
Author(s):  
R. Quackenbush ◽  
C. S. Szabó
Keyword(s):  
Finite Group ◽  
Dihedral Group ◽  
Strong Duality ◽  
Sylow Subgroups ◽  
Metacyclic Groups ◽  
Group D

AbstractDavey and Quackenbush proved a strong duality for each dihedral group Dm with m odd. In this paper we extend this to a strong duality for each finite group with cyclic Sylow subgroups (such groups are known to be metacyclic).

Units in Group Rings of the Infinite Dihedral Group

1991 ◽  
Vol 34 (1) ◽  
pp. 83-89 ◽  
Author(s):  
Maciej Mirowicz
Keyword(s):  
Group Ring ◽  
Integral Domain ◽  
Dihedral Group ◽  
Group Rings ◽  
Finitely Generated ◽  
Group Of Units ◽  
Group D

AbstractThis paper studies the group of units U(RD∞) of the group ring of the infinite dihedral group D∞ over a commutative integral domain R. The structures of U(Z2D∞) and U(Z3D∞) are determined, and it is shown that U(ZD∞) is not finitely generated.

SYMMETRIC BIFURCATIONS IN A RING OF COUPLED DIFFERENTIAL EQUATION WITH PIECEWISE CONTINUOUS ARGUMENTS

2011 ◽  
Vol 21 (02) ◽  
pp. 431-436 ◽  
Author(s):  
BAODONG ZHENG ◽  
JIAN MA ◽  
HUIFENG ZHENG ◽  
CHUNRUI ZHANG
Keyword(s):  
Differential Equation ◽  
Dihedral Group ◽  
Equilibrium Points ◽  
Piecewise Continuous ◽  
Piecewise Continuous Arguments ◽  
Group D

Symmetry bifurcations of equilibrium points of three coupled differential equation with piecewise continuous arguments (EPCA) oscillators are studied. The system is equivariant under dihedral group D3 with order 6. This causes several types of symmetrical bifurcations.

scholarly journals COMMENTS ON DIHEDRAL AND SUPERSYMMETRIC EXTENSIONS OF A FAMILY OF HAMILTONIANS ON A PLANE

2010 ◽  
Vol 25 (27) ◽  
pp. 2373-2380 ◽  
Author(s):  
C. QUESNE
Keyword(s):  
Dihedral Group ◽  
Creation And Annihilation Operators ◽  
Group D ◽  
Trigonometric Identities

For any odd k, a connection is established between the dihedral and supersymmetric extensions of the Tremblay–Turbiner–Winternitz Hamiltonians Hk on a plane. For this purpose, the elements of the dihedral group D2k are realized in terms of two independent pairs of fermionic creation and annihilation operators and some interesting trigonometric identities are demonstrated.

scholarly journals Properties of Suborbits of the Dihedral Group D<sub>n</sub> Acting on Ordered Subsets

2017 ◽  
Vol 07 (08) ◽  
pp. 375-382
Author(s):  
Rose W. Gachogu ◽  
Ireri N. Kamuti ◽  
Moses N. Gichuki
Keyword(s):  
Dihedral Group ◽  
Group D

scholarly journals Skew Spectra of Oriented Graphs

10.37236/270 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Bryan Shader ◽  
Wasin So
Keyword(s):  
Directed Graph ◽  
Adjacency Matrix ◽  
Undirected Graph ◽  
Oriented Graph ◽  
Oriented Graphs ◽  
Underlying Graph ◽  
Adjacency Spectrum ◽  
The Relationship ◽  
Skew Adjacency Matrix

An oriented graph $G^{\sigma}$ is a simple undirected graph $G$ with an orientation $\sigma$, which assigns to each edge a direction so that $G^{\sigma}$ becomes a directed graph. $G$ is called the underlying graph of $G^{\sigma}$, and we denote by $Sp(G)$ the adjacency spectrum of $G$. Skew-adjacency matrix $S( G^{\sigma} )$ of $G^{\sigma}$ is introduced, and its spectrum $Sp_S( G^{\sigma} )$ is called the skew-spectrum of $G^{\sigma}$. The relationship between $Sp_S( G^{\sigma} )$ and $Sp(G)$ is studied. In particular, we prove that (i) $Sp_S( G^{\sigma} ) = {\bf i} Sp(G)$ for some orientation $\sigma$ if and only if $G$ is bipartite, (ii) $Sp_S(G^{\sigma}) = {\bf i} Sp(G)$ for any orientation $\sigma$ if and only if $G$ is a forest, where ${\bf i}=\sqrt{-1}$.

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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