One-regular Normal Cayley Graphs on Dihedral Groups of Valency 4 or 6 with Cyclic Vertex Stabilizer

A graph G is one-regular if its automorphism group Aut（G） acts transitively and semiregularly on the arc set. A Cayley graph Cay（Г, S） is normal if Г is a normal subgroup of the full automorphism group of Cay（Г, S）. Xu, M. Y., Xu, J. （Southeast Asian Bulletin of Math., 25, 355-363 （2001）） classified one-regular Cayley graphs of valency at most 4 on finite abelian groups. Marusic, D., Pisanski, T. （Croat. Chemica Acta, 73, 969-981 （2000）） classified cubic one-regular Cayley graphs on a dihedral group, and all of such graphs turn out to be normal. In this paper, we classify the 4-valent one-regular normal Cayley graphs G on a dihedral group whose vertex stabilizers in Aut（G） are cyclic. A classification of the same kind of graphs of valency 6 is also discussed.

Augmentation quotients for complex representation rings of dihedral groups

Denote by Dm the dihedral group of order 2m. Let R（Dm） be its complex representation ring, and let △（Dm） be its augmentation ideal. In this paper, we determine the isomorphism class of the n-th augmentation quotient △^n（Dm）/△^n＋1（Dm） for each positive integer n.

Enumeration of Cubic Cayley Graphs on Dihedral Groups

Let p be an odd prime, and D2p = （a,b I aP = b2 = l,bab= a 1） the dihedral group of order 2p. In this paper, we completely classify the cubic Cayley graphs on D2p up to isomorphism by means of spectral method. By the way, we show that two cubic Cayley graphs on D2p are isomorphic if and only if they are cospectral. Moreover, we obtain the number of isomorphic classes of cubic Cayley graphs on D2 by using Gauss＇ celebrated law of quadratic reciprocity.

Reliability Analysis of the Cayley Graphs of Dihedral Groups

Cayley graphs have many good properties as models of communication networks. This study analyzes the reliability of the Cayley graph based on the dihedral graph. Graph theory and analyses show that almost all Cayley graphs of the dihedral graph D2n are optimal super-λ. The number Ni（G） of cutsets of size i, λ≤ i≤λ＇ is given as Ni（G） = n[^（n-1）δ i-δ].

Isomorphisms of Cubic Cayley Graphs on Dihedral Groups and Sparse Circulant Matrices

We show that,up to isomorphism,there is a unique non-CI connected cubic Cayley graph on the dihedral group of order 2n for each even number n≥4.This answers in the negative the question of Li whether all connected cubic Cayley graphs are CI-graphs(Discrete Math.,256,301-334(2002)).As an application,a formula is derived for the number of isomorphism classes of connected cubic Cayley graphs on dihedral groups,which generalises the earlier formula of Huang et al.dealing with the particular case when n is a prime(Acta Math.Sin.,Engl.Ser.,33,996-1011(2017)).As another application,a short proof is also given for a result on sparse circulant matrices obtained by Wiedemann and Zieve(arXiv preprint,(2007)).

Coleman Automorphisms of Generalized Dihedral Groups

Let D be a generalized dihedral group and Autcol（D） its Coleman automorphism group. Denote by Outcol（D） the quotient group of Autcol（D） by Inn（D）, where Inn（D） is the inner automorphism group of D. It is proved that either Outcol（D） = i or Outcol（D） is an elementary abelian 2-group whose order is completely determined by the cardinality of π（D）. Furthermore, a necessary and sufficient condition for Outcol（D） = 1 is obtained. In addition, whenever Outcol（D） ≠ 1, it is proved that Autcol（D） is a split extension of Inn（D） by an elementary abelian 2-group for which an explicit description is given.

On the Structure of the Units of Group Algebra of Dihedral Group

In this paper, we completely determine the structure of the unit group of the group algebra of some dihedral groups D2 n over the finite field Fpk, where p is a prime.

Classifying Groups of Small Order

The classification of groups of order less than 16 is reconsidered. The goal of the paper is partly historical and partly pedagogical and aims to achieve the classification as simply as possible in a way which can be easily incorporated into a first course in abstract algebra and without appealing to the Sylow Theorems. The paper concludes with some exercises for students.

Classification of Crystallographic Groups Associated with the Infinite Dihedral Group

Let V be a 2-dimensional vector space over the real field R with an affine or indefinite symmetric bilinear form. The infinite dihedral group W can be viewed as a subgroup of GL(V). In the present paper we will classify all crystallographic groups associated with W up to conjugation in the affine group A(V).

On Fundamental Group of a Certain Class of Welded Knots

In this paper, a certain class of welded knots K;is considered. By calculating the commutators subgroup of fundamental group Gn of welded knot K;,n ∈ Z;, we show that these welded knots are not equivalent to each other and they are all not classical knots. Secondly, we study some properties of Gn and obtain that Gn is linear, residually finite and Hopfian.

Group Orbits of GPS Satellite Configurations for Constellation Management

Air Force Space Command is interested in improving the accuracy of GPS receiver positioning, navigation, and timing. To this end, it is useful to identify a set of optimal satellite constellations where each corresponds to a configuration specifying the number of satellites in each orbital plane. These constellations could then be maintained in a library for future use as satellites fail and are launched. We utilize symmetry in the geometry of the GPS satellite orbits to partition the configurations into a much smaller set of equivalence classes where each class has the same overall receiver accuracy performance. We apply a classical algebraic combinatorial result, Polya＇s Theorem, to count and categorize the classes. Incorporating our results into a GPS constellation optimization computer tool will reduce run time by about an order of magnitude. We apply other algebraic and combinatorial techniques in original ways to count the class sizes and the classes that contain a given number of satellites. Finally, we break the equivalence classes into a still smaller set of new ＂structure＂ classes that are useful in applying the GPS computer tool.

Nowhere-zero 3-flows in Cayley graphs on generalized dihedral group and generalized quaternion group

Tutte conjectured that every 4-edge-connected graph admits a nowhere-zero 3-flow. In this paper, we show that this conjecture is true for Cayley graph on generalized dihedral groups and generalized quaternion groups, which generalizes the result of F. Yang and X. Li [Inform. Process. Lett., 2011, 111： 416-419]. We also generalizes an early result of M. Nanasiova and M. Skoviera [J. Algebraic Combin., 2009, 30： 103-110].

Yetter–Drinfeld Modules over the Hopf–Ore Extension of the Group Algebra of Dihedral Group

Let k be an algebraically closed field of characteristic zero, and Dn be the dihedral group of order 2n, where n is a positive even integer. In this paper, we investigate Yetter-Drinfeld modules over the Hopf-Ore extension A（n,0） of kDn. We describe the structures and properties of simple Yetter Drinfeld modules over A（n, 0）, and classify all simple Yetter-Drinfeld modules over A（n, 0）.

Hermitian Adjacency Spectrum of Cayley Digraphs over Dihedral Group

We first study the spectrum of Hermitian adjacency matrix(H-spectrum)of Cayley digraphs X(D 2n,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.

LCD Codes and Self-orthogonal Codes in Finite Dihedral Group Algebras

Let Fq be a finite field with order q and D2n be the dihedral group with 2n elements, and gcd(q, 2n) = 1. In this article, the authors give precise descriptions and enumerations of linear complementary dual(LCD) codes and self-orthogonal codes in the finite dihedral group algebras Fq[D2n]. Some numerical examples are also presented to illustrate the main results.

Hopf Ore Extension over Dihedral Group

In this paper, the Hopf Ore extension and corresponding module extension of the group algebra over dihedral group are studied. It turns out that the 1-dimensional and 2- dimensional simple representations can both be extended to the simple representations over a class of Hopf Ore extension.

On Non-normal Arc-Transitive 4-Valent Dihedrants

Let X be a connected non-normal 4-valent arc-transitive Cayley graph on a dihedral group Dn such that X is bipartite, with the two bipartition sets being the two orbits of the cyclic subgroup within Dn. It is shown that X is isomorphic either to the lexicographic product Cn[2K1] with n 〉 4 even, or to one of the five sporadic graphs on 10, 14, 26, 28 and 30 vertices, respectively.

On Orientably Regular Non-abelian Covering Maps of the Platonic Maps

We investigate the orientably regular non-abelian coverings of regular maps.A complete classification of dihedral coverings of the Platonic maps for branching over faces(or,dually,vertices)is given.As a result,we generalise the results of Jones and Surowski on regular cyclic coverings of the Platonic maps.