On the complexity of Cayley graphs on a dihedral group

Witte Morris showed in [Discrete Math, 38.1 (1982)] that every connected Cayley graph of a finite (generalized) dihedral group has a Hamiltonian path. The infinite dihedral group is defined as the free product [Formula: see text]. We show that every connected Cayley graph of the free product with amalgamation [Formula: see text] has a Hamiltonian double ray, where [Formula: see text] is a generalized quasi-dihedral group on [Formula: see text] Additionally, this leads to the conclusion that each connected Cayley graph of the infinite dihedral group also contains a Hamiltonian double ray.

The basis of a significant amount of cryptographic systems for information protection are different computationally hard problems. One of these problems is finding the discrete logarithm value in a certain finite group. The problem is to obtain for any two given elements of this group such natural number that the first element to the power of the number equals the second element. In order to implement the cryptosystem, they have to choose an appropriate finite group and an element of high multiplicative order in it, so that computing the discrete logarithm is a hard problem. Powerful quantum computers will solve in polynomial time the discrete logarithm problem in the most common finite groups (multiplicative group of prime or extended finite field, group of points of elliptic curve over a finite field). That is why, as one of directions, they study groups consisting of invertible elements of group rings specified by various rings and groups. In the paper, the issue of finding high order units for special group rings, defined by finite field and dihedral group, is explore

In this work, we investigate isomorphisms of graphs associated with the 216 maximal self-complementary [Formula: see text]-codes over the genetic alphabet [Formula: see text]. Such codes play an important role in maintaining the correct reading frame during the translational process in the ribosome and have been classified into 27 equivalence classes under the action of the dihedral group [Formula: see text]. Naturally, this group action induces graph isomorphisms between the graphs associated with maximal self-complementary [Formula: see text]-codes, as shown in Fimmel etal. (2016). However, we demonstrate here that these induced isomorphisms of the associated graphs are not the only graph isomorphisms between such codes. Specifically, we calculate the largely non-trivial automorphism groups of all the 216 graphs associated to maximal self-complementary [Formula: see text]-codes and we show that no isomorphism exists between maximal self-complementary [Formula: see text]-codes belonging to different equivalence classes. Finally, we provide examples illustrating that the assumptions of maximality, self-complementarity, or the [Formula: see text]-property can not be omitted.

This paper investigates several properties of the Cartesian product of two non-coprime graphs associated with finite groups. Specifically, we focus on key numerical invariants, namely the domination number, independence number, and diameter. The non-coprime graph associated with finite group $G$ is constructed with the vertex set $G\setminus \{e\}$ and connects two distinct vertices if and only if their orders are not coprime. Using this construction, we investigate the Cartesian products of non-coprime graphs associated with various types of groups, including nilpotent groups, dihedral groups, and $p$-groups. We derive several new results, including exact expressions for the domination number, independence number, and diameter of these Cartesian products.

Let G be a group and S be a subset of G in which e /∈ S and S−1 ⊆ S. The Cayley graph of group G with respect to subset S, denoted by Cay(G, S), is an undirected simple graph whose vertices are all elements of G, and two vertices x and y are adjacent if and only if xy−1 ∈ S. If |S| = k, then Cay(G, S) is called a Cayley graph of valency k. The aim of this paper is to determine the structure of Cayley graph of dihedral groups D2n of order 2n when n = p or 2p2, where p is an odd prime number. The graph structures are based on circulant graphs with suitable jumps.

On schurity of dihedral groups

Let [Formula: see text] be a group and [Formula: see text] be the set of all subgroups of [Formula: see text]. The subgroup generating bipartite graph [Formula: see text] defined on [Formula: see text] is a bipartite graph whose vertex set is partitioned into two sets [Formula: see text] and [Formula: see text], and two vertices [Formula: see text] and [Formula: see text] are adjacent if [Formula: see text] is generated by [Formula: see text] and [Formula: see text]. In this paper, we deduce expressions for first and second Zagreb indices of [Formula: see text] and obtain a condition such that [Formula: see text] satisfy [Formula: see text]Hansen–Vukičević conjecture [P. Hansen and D. Vukičević, Comparing the Zagreb indices, Croat. Chem. Acta 80(2) (2007) 165–168]. It is shown that [Formula: see text] satisfies Hansen–Vukičević conjecture if [Formula: see text] is a cyclic group of order [Formula: see text], [Formula: see text] and [Formula: see text]; dihedral group of order [Formula: see text] and [Formula: see text]; and dicyclic group of order [Formula: see text] and [Formula: see text] for any prime [Formula: see text]. While computing Zagreb indices of [Formula: see text] we have computed [Formula: see text] for all [Formula: see text] for the above-mentioned groups. Using these information we also compute Randic Connectivity index, Atom-Bond Connectivity index, Geometric–Arithmetic index, Harmonic index and Sum-Connectivity index of [Formula: see text].

Given a finite group G , the order graph of G, denoted by S(G), is a graph whose vertex set is G, and two distinct vertices a and b are adjacent if o(a) | o(b) or o(b) | o(a), where o(a), and o(b), are the orders of a and b in G, respectively. In this paper, by the order of an element, we give a characterization of the finite groups whose order graph is C4-free. As applications, we classify a few families of finite groups whose order graph is C4-free, such as nilpotent groups, dihedral groups and symmetric groups.

On $k$-invariants of crossed modules corresponding to the dihedral groups

For a real number α, R α ( Γ ) = ∑ h w h x ∈ E ( Γ ) ( deg ( h w ) deg ( h x ) ) α and χ α ( Γ ) = ∑ h w h x ∈ E ( Γ ) ( deg ( h w ) + deg ( h x ) ) α represents the general Randić index and the general Sum-Connectivity index of the graph Γ respectively. In this article, we determine the Randić index, Sum-Connectivity index, the general Randić index, and the general Sum-Connectivity index of the conjugacy class graphs of the Dihedral, Quasi-Dihedral and Generalized Quaternion Groups.

On isomorphisms of tetravalent Cayley digraphs over dihedral groups

Abstract Modern neural networks (NNs) often do not generalize well in the presence of a "covariate shift''; that is, in situations where the training and test data distributions differ, but the conditional distribution of classification labels given the data remains unchanged. In such cases, NN generalization can be reduced to a problem of learning domain-invariant features. Domain adaptation (DA) methods include a broad range of techniques aimed at achieving this; however, these methods have struggled with the need for hyperparameter tuning, which then incurs computational costs. In this work, we introduce SIDDA, an out-of-the-box DA training algorithm built upon the Sinkhorn divergence, that can achieve effective domain alignment with minimal hyperparameter tuning and computational overhead. We demonstrate the efficacy of our method on multiple simulated and real datasets of varying complexity, including simple shapes, handwritten digits, and real astronomical observations. These datasets exhibit covariate shifts due to noise, blurring, and differences between telescopes. SIDDA is compatible with a variety of NN architectures, and it works particularly well in improving classification accuracy and model calibration when paired with equivariant NNs (ENNs). We find that SIDDA enhances the generalization capabilities of NNs, achieving up to a 40% improvement in classification accuracy on unlabeled target data. We also study the efficacy of DA on ENNs with respect to the varying orders of the dihedral group, and find that the model performance improves with higher orders. Finally, we find that SIDDA enhances model calibration on both source and target data, with the most significant gains in the unlabeled target domain---achieving over an order of magnitude improvement in the ECE and Brier score. SIDDA's versatility across various NN models and datasets, combined with its automated approach to domain alignment, has the potential to significantly advance multi-dataset studies by enabling the development of highly generalizable models

According to the Equivariant Hopf Theorem, a consequence of Hopf bifurcation in systems with dihedral [Formula: see text]-symmetry, where [Formula: see text] is the group of symmetries of an N-gon, is for these systems to be able to exhibit, up to conjugacy, three types of responses: a traveling wave form and two standing wave forms. From related work, it is known that, generically, when [Formula: see text], these are the only types of solutions that bifurcate from the trivial solution via a double Hopf bifurcation point. In this paper, we prove, analytically and computationally, that when the rotation symmetry is broken, this double Hopf bifurcation point splits into two standard, back-to-back Hopf bifurcation points, corresponding to the two types of standing wave solutions, but with fixed phase differences. Traveling waves are no longer found to be possible. The results are motivated by the analysis of network systems, in which rotation (or reflection) symmetry can be broken due to the presence of disorder or heterogeneities in system parameters; that is, due to forced symmetry-breaking bifurcations. A brief discussion of ongoing experimental work is also included. This work is expected to have relevance to a number of physical systems.

A theoretical framework is established for explicitly calculating rigid kernels of self-similar regular branch groups. This is applied to a new infinite family of branch groups in order to provide the first examples of self-similar, branch groups with infinite rigid kernel. The groups are analogues of the Hanoi Towers group on 3 pegs, based on the standard actions of finite dihedral groups on regular polygons with odd numbers of vertices, and the rigid kernel is an infinite Cartesian power of the cyclic group of order 2, except for the original Hanoi group. The proofs rely on a symbolic-dynamical approach, related to finitely constrained groups.

For a finite group G, the coprime graph ΓG of G is defined as the graph with vertex set G, the group itself, and two distinct vertices u, v in ΓG are adjacent if and only if gcd(|u|, |v|) = 1, where |u| is the order of u. This study analyzes the characteristic polynomial of matrices for the dihedral group of order 2n, where n is a power of a prime number. In addition, this paper examines the characteristic polynomial of the matrices for a power of a prime number n. The energy of the graph is also obtained.

This paper investigates the primitive and regular characteristics of Dihedral Group of degree 2p, where p is an odd prime. By utilizing numerical approached, the properties of these groups were examined to shed light on their structure, behavior, and underlying algebraic characteristics. The work uses some group concept to test conditions for primitivity and regularity in these groups, with the help of Group Algorithm Programming (GAP) our results were validated. The main focus of this paper is on their applications to musical note theory. We explore the conditions under which these groups exhibit primitive and regular action on sets, highlighting their algebraic properties and symmetries. The theoretical findings are then connected to musical note arrangements, where pitch classes and transformations exhibit similar cyclic and reflective patterns. By establishing this connection, we demonstrate how group-theoretic principles can enhance the understanding of musical scales, chord structures, and symmetrical note sequences. The results presented offer new insights into the intersection of abstract algebra and music, paving the way for further interdisciplinary exploration. The work reveal that the musical note operate base on their pitch classes and musical intervals. It was discovered that the group of transpositions and inversions, denoted T_nT_(1-n) is isomorphic to the dihedral group 12. Finally, Its Conjugacy classes and Character table was presented.

Abstract We obtain restrictions on units of even order in the integral group ring of a finite group by studying their actions on the reductions modulo 4 of lattices over the 2‐adic group ring . This improves the “lattice method” which considers reductions modulo primes , but is of limited use for essentially due to the fact that . Our methods yield results in cases where has blocks, whose defect groups are Klein four groups or dihedral groups of order 8. This allows us to disprove the existence of units of order for almost simple groups with socle where and to answer the prime graph question affirmatively for many such groups.

For a finite group [Formula: see text], and a subset [Formula: see text] of [Formula: see text] such that [Formula: see text], let [Formula: see text] be a Cayley graph of [Formula: see text] relative to [Formula: see text]. An arc-transitive graph [Formula: see text] is said to be 1-regular if [Formula: see text] acts regularly on 1-arcs set. This paper provides a classification of 1-regular normal Cayley graphs on dihedral groups of valency [Formula: see text].

A <em>Γ</em>-supermagic labeling of a graph <em>G</em>=(<em>V</em>,<em>E</em>) is a bijection from <em>E</em> to a group <em>Γ</em> of order |<em>E</em>| such that for every vertex <em>x</em>∈<em>V</em> a product of labels of all edges incident with <em>x</em> is equal to the same element <em>µ</em>∈<em>Γ</em>. A <em>Γ</em>-supermagic labeling of the Cartesian product of two cycles, <em>C</em><sub>m</sub>ℹ<em>C</em><sub>n</sub> for every <em>m</em>,<em>n</em>≥3 of the same parity was found recently [5, 6] for all Abelian groups of order 2<em>mn</em>. In this paper we present a <em>D</em><sub>k</sub>-supermagic labeling of the Cartesian, direct, and strong product by dihedral group <em>D</em><sub>k</sub> for any <em>m</em>,<em>n</em>≥3.