Semidihedral Groups Research Articles

Smooth skew morphisms on semi-dihedral groups

Smooth skew morphisms on semi-dihedral groups

Braces with adjoint group of maximal nilpotency class

Computer calculations (Guarnieri and Vendramin, 2017) suggested that the number of braces associated with 2-groups of maximal nilpotency class stabilizes with increasing order. The conjectured seven braces for a generalized quaternion group of order ⩾32 have been classified by the second author (J. Group Theory, 2020). A similar result for dihedral 2-groups was obtained recently by Byott and Ferri. The remaining open case of semidihedral groups is accomplished in the paper.

Laplacian Spectrum and Energy of the Cyclic Order Product Prime Graph of Semi-dihedral Groups

This paper introduces cyclic order product prime graphs to examine the spectral graph properties of semi-dihedral groups, specifically focusing on the Laplacian spectrum and energy. Combining principles from cyclic graphs and order product prime graphs enhances understanding of group algebraic structures. In this context, the cyclic order product prime graph for a finite group is defined as one where two distinct vertices are adjacent if their generated subgroup is cyclic, and the product of their orders is a prime power. Our methodological approach begins with establishing a general presentation for these graphs within semi-dihedral groups. This foundational step is essential for deriving some properties, such as vertex degrees, the number of edges, and Laplacian characteristic polynomials. This information subsequently facilitates the determination of the Laplacian spectrum, characterized by seven eigenvalues of various multiplicities, and the computation of their Laplacian energy. The semi-dihedral group of order 16 is a particular example to illustrate the practicality and generality of our theorems.

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.

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.

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.

Constructing directed strongly regular graphs by using semidirect products and semidihedral groups

In this paper, directed strongly regular graphs (DSRGs) are constructed by using semidirect products. The orbit condition in [3] has been weakened and this gives rise to the construction of DSRGs. Moreover, a different construction is given for DSRG by using semidihedral groups.

Hitting time, resistance distance and Kirchhoff index of Cayley graphs over dicyclic and semi-dihedral groups

In this paper, using eigenvalues and eigenvectors, we derive the closed-form formulae for the hitting time, resistance distance and Kirchhoff index of the Cayley graphs over dicyclic and semi-dihedral groups.

Perfect state transfer on bi-Cayley graphs over abelian groups

The study of perfect state transfer on graphs has attracted a great deal of attention during the past ten years because of its applications to quantum information processing and quantum computation. Perfect state transfer is understood to be a rare phenomenon. This paper establishes necessary and sufficient conditions for a bi-Cayley graph having a perfect state transfer over any given finite abelian group. As corollaries, many known and new results are obtained on Cayley graphs having perfect state transfer over abelian groups, (generalized) dihedral groups, semi-dihedral groups and generalized quaternion groups. Especially, we give an example of a connected non-normal Cayley graph over a dihedral group having perfect state transfer between two distinct vertices, which was thought impossible.

Metric Properties of Non-Commuting Graph Associated to Two Groups

The non-commuting graph associated to a group has non-central elements of the graph as vertices and two elements [Formula: see text] and [Formula: see text] do not form an edge if and only if [Formula: see text]. In this paper, we consider non-commuting graphs associated to dihedral and semidihedral groups. We investigate their metric properties such as center, periphery, eccentric graph, closure and interior. We also perform various types of metric identifications on these graphs. Moreover, we generate metric and metric-degree polynomials of these graphs.

Mostar index of graphs associated to groups

Abstract A bond-additive connectivity index, named as the Mostar index, is used to measure the amount of peripheral edges of a simple connected graph, where a peripheral edge in a graph is an edge whose one end vertex has more number of vertices closer as compared to the other end vertex. In this study, we count the contribution of peripheral edges in commuting, non-commuting, and non-conjugate graphs associated to the dihedral and semi-dihedral groups. In fact, we compute the Mostar index of these graphs.

Perfect codes and universal adjacency spectra of commuting graphs of finite groups

The commuting graph [Formula: see text] of a finite group [Formula: see text] has vertex set as [Formula: see text], and any two distinct vertices [Formula: see text] are adjacent if [Formula: see text] and [Formula: see text] commute with each other. In this paper, we first study the perfect codes of [Formula: see text]. We then find the universal adjacency spectra of the join of two regular graphs, join of two regular graphs in which one graph is a union of two regular graphs, and generalized join of regular graphs in terms of adjacency spectra of the constituent graphs and an auxiliary matrix. As a consequence, we obtain the adjacency, Laplacian, signless Laplacian, and Seidel spectra of the above graph operations. As an application of the results obtained, we calculate the adjacency, Laplacian, signless Laplacian, and Seidel spectra of [Formula: see text] for [Formula: see text], where [Formula: see text] is the dihedral group, [Formula: see text] is the dicyclic group and [Formula: see text] is the semidihedral group. Moreover, we provide the exact value of the spectral radius of the adjacency, Laplacian, signless Laplacian, and Seidel matrix of [Formula: see text] for [Formula: see text]. Some of the theorems published in [F. Ali and Y. Li, The connectivity and the spectral radius of commuting graphs on certain finite groups, Linear Multilinear Algebra 69 (2019) 1–14; T. Cheng, M. Dehmer, F. Emmert-Streib, Y. Li and W. Liu, Properties of commuting graphs over semidihedral groups, Symmetry 13(1) (2021) 103] can be deduced as corollaries from the theorems obtained in this paper.

Generalized Commuting Graph of Dihedral, Semi-dihedral and Quasi-dihedral Groups

Commuting graphs are characterized by vertices that are non-central elements of a group where two vertices are adjacent when they commute. In this paper, the concept of commuting graph is extended by defining the generalized commuting graph. Furthermore, the generalized commuting graph of the dihedral groups, the quasi-dihedral groups and the semi-dihedral groups are presented and discussed. The graph properties including chromatic and clique numbers are also explored.

Perfect quantum state transfer on Cayley graphs over semi-dihedral groups

Perfect quantum state transfer plays a crucial role in quantum information processing and quantum computation. There has been extensive study of perfect quantum state transfer on Cayley graphs over abelian groups. In this paper, we consider the existence of perfect quantum state transfer on Cayley graphs over semi-dihedral groups which are non-abelian groups. Using the representations of semi-dihedral groups, we provide some necessary and sufficient conditions for Cayley graphs over semi-dihedral groups admitting perfect quantum state transfer. By those conditions, we present examples of perfect quantum state transfer on Cayley graphs over semi-dihedral groups. In addition, we propose results about whether some new Cayley graphs over non-abelian groups admit perfect quantum state transfer.

Pretty good state transfer on Cayley graphs over semi-dihedral groups

Let Γ be a graph with adjacency matrix A. The transition matrix of Γ corresponding to A is defined by , where and . The graph is said to exhibit pretty good state transfer between a pair of vertices u and v if there exists a sequence of real numbers and a complex number γ with unit norm such that In this paper, we explore pretty good state transfer on Cayley graphs over semi-dihedral groups by using the representations of such groups. We show that graphs have pretty good state transfer for some suitable subsets S if n is a power of 2. Moreover, we present a sufficient and necessary condition for a non-integral graph Cay to admit pretty good state transfer. Some concrete constructions of Cayley graphs over semi-dihedral groups having PGST are also presented.

Integral Cayley graphs over semi-dihedral groups

Classifying integral graphs is a hard problem that initiated by Harary and Schwenk in 1974. In this paper, with the help of character table, we treat the corresponding problem for Cayley graphs over the semi-dihedral group SD8n = ?a,b | a4n = b2 = 1, bab = a2n-1?, n ? 2. We present several necessary and sufficient conditions for the integrality of Cayley graphs over SD8n, we also obtain some simple sufficient conditions for the integrality of Cayley graphs over SD8n in terms of the Boolean algebra of hai. In particular, we give the sufficient conditions for the integrality of Cayley graphs over semi-dihedral groups SD2n (n?4) and SD8p for a prime p, from which we determine several infinite classes of integral Cayley graphs over SD2n and SD8p.

The maximal class 2-groups generate the same variety

In this note, we derive a criterion for determining when two groups generate the same group variety. We use this, along with a description of the 2-variable word maps on the dihedral, dicyclic, and semi-dihedral groups, to show that the maximal class 2-groups of the same order generate the same variety.

Properties of Commuting Graphs over Semidihedral Groups

This paper considers commuting graphs over the semidihedral group SD8n. We compute their eigenvalues and obtain that these commuting graphs are not hyperenergetic for odd n≥15 or even n≥2. We further compute the Laplacian spectrum, the Laplacian energy and the number of spanning trees of the commuting graphs over SD8n. We also discuss vertex connectivity, planarity, and minimum disconnecting sets of these graphs and prove that these commuting graphs are not Hamiltonian.

Counting the number of supercharacter theories of a finite group

The supercharacter theory of a finite group is a generalization of the ordinary character theory of finite groups that was introduced by Diaconis and Isaacs in 2008. In this paper, the concept of groups with quasi-identical character tables are presented. It is proved that the groups with quasi-identical character tables have the same number of supercharacter theories. As a consequence, the dihedral and semi-dihedral groups of order 2n, n≥3, have the same number of supercharacter theories.

Enumerating regular graph coverings whose covering transformation groups are ℤ_2-extensions of a cyclic group

Several types of the isomorphism classes of graph coverings have been enumerated by many authors. In 1988, Hofmeister enumerated the double covers of a graph, and this work was extended to n -fold coverings of a graph by the second and third authors. For regular coverings of a graph, their isomorphism classes were enumerated when the covering transformation group is a finite abelian or dihedral group. In this paper, we enumerate the isomorphism classes of graph coverings when the covering transformation group is a ℤ 2 -extension of a cyclic group, including generalized quaternion and semi-dihedral groups.