Dicyclic Groups Research Articles

Fractional revival on semi-Cayley graphs over abelian groups

In this paper, we investigate the existence of fractional revival on semi-Cayley graphs over finite abelian groups. We give some necessary and sufficient conditions for semi-Cayley graphs over finite abelian groups admitting fractional revival. We also show that integrality is necessary for some semi-Cayley graphs admitting fractional revival. Moreover, we characterize the minimum time when semi-Cayley graphs admit fractional revival. As applications, we give examples of certain Cayley graphs over the generalized dihedral groups and generalized dicyclic groups admitting fractional revival.

Ground states of Class S theory on ADE singularities and dual Chern-Simons theory

In radial quantization, the ground states of a gauge theory on ADE singularities ℝ4/Γ are characterized by flat connections that are maps from Γ to the gauge group. We study Class S theory of type a1 = su2 on a Riemann surface of genus g > 1, without punctures. The fundamental building block of Class S theory is the trifundamental Trinion theory — a low energy limit of two M5 branes compactified on the three-punctured Riemann sphere. We show, through the superconformal index, that the supersymmetric Casimir energy of the trifundamental theory imposes a constraint on the set of allowed flat connections, which agrees with the prediction of a duality relating the ground state Hilbert space of Class S on ADE singularities to the Hilbert space of a certain dual Chern-Simons theory whose gauge group is given by the McKay correspondence. The conjecture is shown to hold for Γ = ℤk, agreeing with the previous results of Benini et al. and Alday et al. A non-abelian generalization of this duality is analyzed by considering the example of the dicyclic group Γ = Dic2, corresponding to Chern-Simons gauge group SO(8).

Minimal directed strongly regular Cayley graphs over generalized dicyclic groups

Minimal directed strongly regular Cayley graphs over generalized dicyclic groups

Normal Supercharacter Theories Of The Dicyclic Groups

The supercharacter theory was developed by P. Diaconis and I. M. Isaacs as a natural generalization of the classical ordinary character theory. There are different constructions for finding the supercharacter theories of a finite group. Supercharacter theories of many finite groups, such as cyclic groups, Frobenius groups, dihedral groups, elementary abelian $p$-groups, and Camina groups, etc. are studied with different constructions. One of the constructions uses normal subgroups. In this paper, we consider dicyclic groups and find some of their normal supercharacter theories and some automorphic supercharacter theories in special cases.

A new class of directed strongly regular Cayley graphs over dicyclic groups

<p>We endeavored to investigate directed strongly regular Cayley graphs (or DSRCG for short) over dicyclic groups $ \operatorname{Dic}_{4n} = \langle \alpha, \beta\; |\; \alpha^{n} = \beta^4 = 1, \beta^{-1}\alpha\beta = \alpha^{-1}\rangle $, where $ n $ is odd. We derived several DSRCGs over $ \operatorname{Dic}_{4n} $ for $ n $ odd. We then derived a criterion for a certain class of Cayley graph to be directed strongly regular.</p>

Lee metrics on groups

In this work we consider interval metrics on groups; that is, integral invariant metrics whose associated weight functions do not have gaps. We give conditions for a group to have and not to have interval metrics. Then we study Lee metrics on general groups, that is interval metrics having the finest unitary symmetric associated partition. These metrics generalize the classic Lee metric on cyclic groups. In the case that $ G $ is a torsion-free group or a finite group of odd order, we prove that $ G $ has a Lee metric if and only if $ G $ is cyclic. Also, if $ G $ is a group admitting Lee metrics then $ G \times \mathbb{Z}_2^k $ always has Lee metrics for every $ k \in \mathbb{N} $. Then, we show that some families of metacyclic groups, such as cyclic, dihedral, and dicyclic groups, always have Lee metrics. Finally, we give conditions for non-cyclic groups such that they do not have Lee metrics. We end with tables of all groups of order $ \le 31 $ indicating which of them have (or have not) Lee metrics and why (not).

Acentralizers of some Finite Groups

Let G be a finite group. The acentralizer of an automorphism α of G, is the subgroup of fixed points of α, i.e., CG(α) = {g ∈ G∣α(g) = g}. In this paper we determine the acentralizers of the dihedral group of order 2n, the dicyclic group of order 4n and the symmetric group on n letters. As a result we see that if n ≥ 3, then the number of acentralizers of the dihedral group and the dicyclic group of order 4n are equal. Also we determine the acentralizers of groups of orders pq and pqr, where p, q and r are distinct primes.

Algebraic degree of Cayley graphs over dicyclic and semi-dihedral groups

We explore the splitting field F(X) for a simple connected graph X, which is the smallest field extension of Q that encompasses all eigenvalues of a specific adjacency matrix associated with X. The algebraic degree of X, denoted as [F(X):Q], represents the extension degree of this field. Our study focuses on deriving both upper and lower bounds for the algebraic degrees of Cayley graphs over the dicyclic group and the semi-dihedral group. Furthermore, we provide detailed analysis on the algebraic degrees and the corresponding splitting fields for normal mixed Cayley graphs over these two groups.

Sumsets in dicyclic groups Q 4 n and Um,n

Let G be an arbitrary group. For any subsets A and B of G, let A * B = { a * b ; a ∈ A , b ∈ B } where ` * ’ is the binary operation on G. By μ G ( r , s ) , we denote the minimum cardinality of the set A * B , where | A | = r and | B | = s . In 2003, Eliahou et al. proposed a conjecture that for every finite group G of order g and every pair of integers r , s with 1 ≤ r , s ≤ g , k G ( r , s ) ≤ μ G ( r , s ) , where k G ( r , s ) = min d ∈ H ( G ) { ( ⌈ r d ⌉ + ⌈ s d ⌉ − 1 ) d } and H ( G ) is the set of orders of finite subgroups of G. In 2003, Eliahou et al. verified the conjecture for dihedral group D q of index q = p v . In this paper, we determine the upper bound for the size of μ Q 4 n ( r , s ) and μ U m , n ( r , s ) , where Q 4 n = 〈 a , b : a 2 n = e , b 2 = a n , a b = b a − 1 〉 is a dicyclic group of order 4n and U m , n = 〈 a , b : a 2 n = e , b m = e , a b a − 1 = b − 1 〉 and verify the conjecture proposed by Eliahou et al. [7] for Q 4 n for n to be a prime power.

Algebraic degrees of 2-Cayley digraphs over abelian groups

A digraph Γ is called a 2-Cayley digraph over a group G if there exists a 2-orbit semiregular subgroup of Aut(Γ) isomorphic to G. In this paper, we give upper and lower bounds for the algebraic degrees of 2-Cayley digraphs over abelian groups. As an application, we consider the Cayley digraphs over finite groups admitting an abelian subgroup of index 2. Special attention is paid for Cayley graphs over generalized dihedral groups and generalized dicyclic groups. For Cayley graphs over generalized dihedral groups and generalized dicyclic groups, we completely determine their algebraic degrees and splitting fields. At last, the upper and lower bounds for the algebraic degrees of Cayley digraphs over semi-dihedral groups are obtained.

On (distance) Laplacian characteristic polynomials of power graphs

The characteristic polynomials of the Laplacian and the distance Laplacian matrices of power graphs of groups of order [Formula: see text], where [Formula: see text] and [Formula: see text] are primes, are obtained. Further, the characteristic polynomials of these matrices for proper power graphs of cyclic and dicyclic groups are given. The important inequalities for the zeros of the distance Laplacian characteristic polynomials of power graphs of finite groups are presented in comments.

Supercharacter theories of the dicyclic groups

In this paper, all supercharacter theories of dicyclic groups are described. These supercharacter theories are constructed by using the supercharacter theories of dihedral groups of order 4n which are classified by Lamar.

On integral mixed Cayley graphs over non-abelian finite groups admitting an abelian subgroup of index 2

Recently, several works by a number of authors have provided characterizations of integral undirected Cayley graphs over generalized dihedral groups and generalized dicyclic groups. We generalize and unify these results in two different ways. Firstly, we work over arbitrary non-abelian finite groups admitting an abelian subgroup of index 2. Secondly, our main result actually characterizes integral mixed Cayley graphs over such finite groups, in the spirit of a very recent result of Kadyan–Bhattarcharjya in the abelian case.

A classification of tetravalent connected vertex-transitive bi-dicirculants

A graph is said to be a bi-Cayley graph over a group H if it admits H as a group of automorphisms acting semiregularly on its vertex-set with two orbits of equal size. A bi-Cayley graph over a dicyclic group is called a bi-dicirculant. We give a classification of tetravalent connected vertex-transitive bi-dicirculants in this paper. As a byproduct, we prove that all connected tetravalent vertex-transitive bi-dicirculants are Cayley graphs.

Descending endomorphism graphs of groups

We define a new type of graph of a group with reference to the descending endomorphisms of the group. A descending endomorphism of a group is an endomorphism that induces a corresponding endomorphism in every homomorphic image of the group. We define the undirected (directed) descending endomorphism graph of a group as the undirected (directed) graph whose vertex set is the underlying set of the group, in which there is an undirected (directed) edge from one vertex to another if the group has a descending endomorphism that maps the former element to the latter. We investigate some basic properties of these graphs and show that they are closely related to power graphs. We also determine the descending endomorphism graphs of symmetric, dihedral, and dicyclic groups.

Perfect state transfer on weighted bi-Cayley graphs over abelian groups

The study of perfect state transfer on graphs has attracted extensive attention because of its application in quantum information and computation. This paper establishes necessary and sufficient conditions for a weighted bi-Cayley graph having perfect state transfer over any given finite abelian group. As special cases of weighted bi-Cayley graphs, the results in this paper can apply to examine weighted Cayley graphs having perfect state transfer over dihedral groups, generalized dihedral groups, semi-dihedral groups and dicyclic groups.

Distance-regular Cayley graphs over dicyclic groups

The characterization of distance-regular Cayley graphs originates from the problem of identifying strongly regular Cayley graphs, or equivalently, regular partial difference sets. In this paper, a partial classification of distance-regular Cayley graphs on dicyclic groups is obtained. More specifically, it is shown that every distance-regular Cayley graph on a dicyclic group is a complete graph, a complete multipartite graph, or a non-antipodal bipartite distance-regular graph with diameter 3 satisfying some additional conditions.

Perfect quantum state transfer on Cayley graphs over dicyclic groups

ABSTRACT Perfect state transfer has attracted much interest recently due to its significant applications in quantum information processing and cryptography. In this paper, we investigate perfect state transfer on Cayley graphs over dicyclic groups. Utilizing the representations and characters of dicyclic groups , some necessary and sufficient conditions for Cayley graphs over dicyclic groups admitting perfect state transfer are provided. By those results, some concrete constructions are presented.

The extreme vertices of the power graph of a group

For a fixed finite group G, the power graph of G was defined to be the simple graph Γ(G) whose vertex set V(Γ(G))=G, and edge set E(Γ(G))={xy: either x=yn or y=xn for some integer n}. In this paper the extreme vertices of the power graph of abelian groups, dihedral groups and dicyclic groups have been characterized.

Formula omitted]-property of some groups

In this paper, firstly, we determine the local 2-GCI-property and 2−GCI-property of the cyclic group. Then, for the dihedral group D2n, we prove that it has local GCI-property if and only if n is an odd prime or 9. Further, the dihedral group D2n cannot have GCI-property. Moreover, we discuss the GCI-property of the elementary abelian group, the dicyclic group and the semi-dihedral group.