scholarly journals On Minimal Distance-Regular Cayley Graphs of Generalized Dihedral Groups

10.37236/9755 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Štefko Miklavič ◽  
Primož Šparl
Keyword(s):  
Closed Subset ◽  
Dihedral Group ◽  
Cayley Graphs ◽  
Complete Classification ◽  
Minimal Distance ◽  
Dihedral Groups ◽  
Generalized Dihedral Group

Let $G$ denote a finite generalized dihedral group with identity $1$ and let $S$ denote an inverse-closed subset of $G \setminus \{1\}$, which generates $G$ and for which there exists $s \in S$, such that $\langle S \setminus \{s,s^{-1}\} \rangle \ne G$. In this paper we obtain the complete classification of distance-regular Cayley graphs $\mathrm{Cay}(G;S)$ for such pairs of $G$ and $S$.

Pentavalent One-regular Graphs of Square-free Order

2010 ◽  
Vol 17 (03) ◽  
pp. 515-524 ◽  
Author(s):  
Yantao Li ◽  
Yan-Quan Feng
Keyword(s):  
Automorphism Group ◽  
Regular Graph ◽  
Cayley Graphs ◽  
Regular Graphs ◽  
Dihedral Groups ◽  
Free Order

A graph is one-regular if its automorphism group acts regularly on the set of its arcs. Let n be a square-free integer. It is shown in this paper that a pentavalent one-regular graph of order n exists if and only if n = 2 · 5tp1p2 … ps ≥ 62, where t ≤ 1, s ≥ 1, and pi's are distinct primes such that 5|(pi-1). For such an integer n, there are exactly 4s-1 non-isomorphic pentavalent one-regular graphs of order n, which are Cayley graphs on dihedral groups constructed by Kwak et al. This work is a continuation of the classification of cubic one-regular graphs of order twice a square-free integer given by Zhou and Feng.

Isomorphisms of bi-Cayley graphs on Dihedral groups

2020 ◽  
Vol 12 (04) ◽  
pp. 2050051
Author(s):  
Majid Arezoomand ◽  
Afshin Behmaram ◽  
Mohsen Ghasemi ◽  
Parivash Raeighasht
Keyword(s):  
Bipartite Graph ◽  
Cayley Graph ◽  
Dihedral Group ◽  
Cayley Graphs ◽  
Dihedral Groups ◽  
Vertex Set

For a group [Formula: see text] and a subset [Formula: see text] of [Formula: see text] the bi-Cayley graph BCay[Formula: see text] of [Formula: see text] with respect to [Formula: see text] is the bipartite graph with vertex set [Formula: see text] and edge set [Formula: see text]. A bi-Cayley graph BCay[Formula: see text] is called a BCI-graph if for any bi-Cayley graph BCay[Formula: see text], [Formula: see text] implies that [Formula: see text] for some [Formula: see text] and [Formula: see text]. A group [Formula: see text] is called a [Formula: see text]-BCI-group if all bi-Cayley graphs of [Formula: see text] with valency at most [Formula: see text] are BCI-graphs. In this paper, we characterize the [Formula: see text]-BCI dihedral groups for [Formula: see text]. Also, we show that the dihedral group [Formula: see text] ([Formula: see text] is prime) is a [Formula: see text]-BCI-group.

scholarly journals A Complete Classification of Which (n, k)-Star Graphs are Cayley Graphs

2017 ◽  
Vol 34 (1) ◽  
pp. 241-260
Author(s):  
Karimah Sweet ◽  
Li Li ◽  
Eddie Cheng ◽  
László Lipták ◽  
Daniel E. Steffy
Keyword(s):  
Cayley Graphs ◽  
Complete Classification ◽  
Star Graphs

scholarly journals Integral cayley graphs over semi-dihedral groups

2021 ◽  
pp. 1-1
Author(s):  
Tao Cheng ◽  
Lihua Feng ◽  
Guihai Yu ◽  
Chi Zhang
Keyword(s):  
Boolean Algebra ◽  
Dihedral Group ◽  
Cayley Graphs ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Character Table ◽  
Hard Problem ◽  
Dihedral Groups ◽  
Necessary And Sufficient ◽  
Integral Graphs

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.

scholarly journals The Energy of Cayley Graphs for a Generating Subset of the Dihedral Groups

MATEMATIKA ◽  
2019 ◽  
Vol 35 (3) ◽  
Author(s):  
Amira Fadina Ahmad Fadzil ◽  
Nor Haniza Sarmin ◽  
Ahmad Erfanian
Keyword(s):  
Finite Group ◽  
Cayley Graph ◽  
Adjacency Matrix ◽  
Dihedral Group ◽  
Cayley Graphs ◽  
Dihedral Groups ◽  
Specific Subset ◽  
A Finite Group

Let G be a finite group and S be a subset of G where S does not include the identity of G and is inverse closed. A Cayley graph of a group G with respect to the subset S is a graph where its vertices are the elements of G and two vertices a and b are connected if ab^(−1) is in the subset S. The energy of a Cayley graph is the sum of all absolute values of the eigenvalues of its adjacency matrix. In this paper, we consider a specific subset S = {b, ab, . . . , a^(n−1)b} for dihedral group of order 2n, where n is greater or equal to 3 and find the Cayley graph with respect to the set. We also calculate the eigenvalues and compute the energy of the respected Cayley graphs. Finally, the generalization of the energy of the respected Cayley graphs is found.

scholarly journals The structure of Cayley graphs of dihedral groups of Valencies 1, 2 and 3

2021 ◽  
Vol 40 (6) ◽  
pp. 1683-1691
Author(s):  
Saba AL-Kaseasbeh ◽  
Ahmad Erfanian
Keyword(s):  
Cayley Graph ◽  
Dihedral Group ◽  
Cayley Graphs ◽  
Dihedral Groups

Let G be a group and S be a subset of G such that e ∉ S and S−1 ⊆ S. Then Cay(G, S) is a simple undirected Cayley graph whose vertices are all elements of G and two vertices x and y are adjacent if and only if xy−1 ∈ S. The size of subset S is called the valency of Cay(G, S). In this paper, we determined the structure of all Cay(D2n, S), where D2n is a dihedral group of order 2n, n ≥ 3 and |S| = 1, 2 or 3.

scholarly journals Multiplicative automatic sequences

2021 ◽  
Author(s):  
Jakub Konieczny ◽  
Mariusz Lemańczyk ◽  
Clemens Müllner
Keyword(s):  
Complete Classification ◽  
Automatic Sequences ◽  
Complex Valued

AbstractWe obtain a complete classification of complex-valued sequences which are both multiplicative and automatic.

scholarly journals Characteristic cohomology and observables in higher spin gravity

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Alexey Sharapov ◽  
Evgeny Skvortsov
Keyword(s):  
Higher Spin Gravity ◽  
Complete Classification ◽  
Gravity Models ◽  
Simons Theory ◽  
Surface Charges ◽  
Higher Spin ◽  
Dynamical Invariants ◽  
Chern Simons ◽  
Conserved Currents

Abstract We give a complete classification of dynamical invariants in 3d and 4d Higher Spin Gravity models, with some comments on arbitrary d. These include holographic correlation functions, interaction vertices, on-shell actions, conserved currents, surface charges, and some others. Surprisingly, there are a good many conserved p-form currents with various p. The last fact, being in tension with ‘no nontrivial conserved currents in quantum gravity’ and similar statements, gives an indication of hidden integrability of the models. Our results rely on a systematic computation of Hochschild, cyclic, and Chevalley-Eilenberg cohomology for the corresponding higher spin algebras. A new invariant in Chern-Simons theory with the Weyl algebra as gauge algebra is also presented.

scholarly journals Nil-clean quadratic elements

2017 ◽  
Vol 16 (10) ◽  
pp. 1750197 ◽  
Author(s):  
Janez Šter
Keyword(s):  
Complete Classification ◽  
Strong Condition ◽  
Clean Rings

We provide a strong condition holding for nil-clean quadratic elements in any ring. In particular, our result implies that every nil-clean involution in a ring is unipotent. As a consequence, we give a complete classification of weakly nil-clean rings introduced recently in [Breaz, Danchev and Zhou, Rings in which every element is either a sum or a difference of a nilpotent and an idempotent, J. Algebra Appl. 15 (2016) 1650148, doi: 10.1142/S0219498816501486].

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