The girth of Cayley graphs over Sylow 2-subgroups of the symmetric groups S2n with diagonal bases

2019 ◽  
Vol 18 (12) ◽  
pp. 1950237
Author(s):  
Bartłomiej Pawlik
Keyword(s):  
Symmetric Group ◽  
Cayley Graph ◽  
Cayley Graphs ◽  
Symmetric Groups ◽  
Polynomial Representation ◽  
Generating Set ◽  
Minimal Generating Set

A diagonal base of a Sylow 2-subgroup [Formula: see text] of symmetric group [Formula: see text] is a minimal generating set of this subgroup consisting of elements with only one nonzero coordinate in the polynomial representation. For different diagonal bases, Cayley graphs over [Formula: see text] may have different girths (i.e. minimal lengths of cycles). In this paper, all possible values of girths of Cayley graphs over [Formula: see text] with diagonal bases are calculated. A criterion for whenever such Cayley graph has girth equal to 4 is presented.

Integral Cayley Graphs over Finite Groups

2020 ◽  
Vol 27 (01) ◽  
pp. 131-136
Author(s):  
Elena V. Konstantinova ◽  
Daria Lytkina
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Cayley Graph ◽  
Cyclic Group ◽  
Finite Groups ◽  
Cayley Graphs ◽  
Alternating Group ◽  
Generating Set ◽  
A Finite Group

We prove that the spectrum of a Cayley graph over a finite group with a normal generating set S containing with every its element s all generators of the cyclic group 〈s〉 is integral. In particular, a Cayley graph of a 2-group generated by a normal set of involutions is integral. We prove that a Cayley graph over the symmetric group of degree n no less than 2 generated by all transpositions is integral. We find the spectrum of a Cayley graph over the alternating group of degree n no less than 4 with a generating set of 3-cycles of the form (k i j) with fixed k, as {−n+1, 1−n+1, 22 −n+1, …, (n−1)2 −n+1}.

scholarly journals Automorphism Groups of a Class of Cubic Cayley Graphs on Symmetric Groups

2017 ◽  
Vol 24 (04) ◽  
pp. 541-550
Author(s):  
Xueyi Huang ◽  
Qiongxiang Huang ◽  
Lu Lu
Keyword(s):  
Automorphism Group ◽  
Symmetric Group ◽  
Cayley Graph ◽  
Automorphism Groups ◽  
Regular Representation ◽  
Cayley Graphs ◽  
Symmetric Groups ◽  
Full Automorphism Group ◽  
Normal Cayley Graph ◽  
The Right

Let Sndenote the symmetric group of degree n with n ≥ 3, S = { cn= (1 2 ⋯ n), [Formula: see text], (1 2)} and Γn= Cay(Sn, S) be the Cayley graph on Snwith respect to S. In this paper, we show that Γn(n ≥ 13) is a normal Cayley graph, and that the full automorphism group of Γnis equal to Aut(Γn) = R(Sn) ⋊ 〈Inn(ϕ) ≅ Sn× ℤ2, where R(Sn) is the right regular representation of Sn, ϕ = (1 2)(3 n)(4 n−1)(5 n−2) ⋯ (∊ Sn), and Inn(ϕ) is the inner isomorphism of Sninduced by ϕ.

scholarly journals The Energy of Cayley Graphs for Symmetric Groups of Order 24

2020 ◽  
pp. 1-6
Author(s):  
Amira Fadina Ahmad Fadzil ◽  
Nor Haniza Sarmin ◽  
Ahmad Erfanian
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Cayley Graph ◽  
Adjacency Matrix ◽  
Cayley Graphs ◽  
Symmetric Groups ◽  
Finite Graph ◽  
Energy Of Graph ◽  
A Finite Group

A Cayley graph of a finite group G with respect to a subset S of G is a graph where the vertices of the graph are the elements of the group and two distinct vertices x and y are adjacent to each other if xy−1 is in the subset S. The subset of the Cayley graph is inverse closed and does not include the identity of the group. For a simple finite graph, the energy of a graph can be determined by summing up the positive values of the eigenvalues of the adjacency matrix of the graph. In this paper, the graph being studied is the Cayley graph of symmetric group of order 24 where S is the subset of S4 of valency up to two. From the Cayley graphs, the eigenvalues are calculated by constructing the adjacency matrix of the graphs and by using some properties of special graphs. Finally, the energy of the respected Cayley graphs is computed and presented. Keywords: energy of graph; cayley graph; symmetric groups

scholarly journals On the density of Cayley graphs of R.Thompson’s group F in symmetric generators

2021 ◽  
pp. 1-13
Author(s):  
V. S. Guba
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
Finite Graph ◽  
Vertex Degree ◽  
Doubling Property ◽  
Generating Set ◽  
Finite Set ◽  
Standard Set ◽  
Thompson’S Group ◽  
Average Vertex

By the density of a finite graph we mean its average vertex degree. For an [Formula: see text]-generated group, the density of its Cayley graph in a given set of generators, is the supremum of densities taken over all its finite subgraphs. It is known that a group with [Formula: see text] generators is amenable if and only if the density of the corresponding Cayley graph equals [Formula: see text]. A famous problem on the amenability of R. Thompson’s group [Formula: see text] is still open. Due to the result of Belk and Brown, it is known that the density of its Cayley graph in the standard set of group generators [Formula: see text], is at least [Formula: see text]. This estimate has not been exceeded so far. For the set of symmetric generators [Formula: see text], where [Formula: see text], the same example only gave an estimate of [Formula: see text]. There was a conjecture that for this generating set equality holds. If so, [Formula: see text] would be non-amenable, and the symmetric generating set would have the doubling property. This would mean that for any finite set [Formula: see text], the inequality [Formula: see text] holds. In this paper, we disprove this conjecture showing that the density of the Cayley graph of [Formula: see text] in symmetric generators [Formula: see text] strictly exceeds [Formula: see text]. Moreover, we show that even larger generating set [Formula: see text] does not have doubling property.

scholarly journals Random spanning trees of Cayley graphs and an associated compactification of semigroups

1999 ◽  
Vol 42 (3) ◽  
pp. 611-620
Author(s):  
Steven N. Evans
Keyword(s):  
Random Walk ◽  
Cayley Graph ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Cayley Graphs ◽  
Finitely Generated ◽  
Finite Tree ◽  
Generating Set ◽  
Countably Infinite

A sequential construction of a random spanning tree for the Cayley graph of a finitely generated, countably infinite subsemigroup V of a group G is considered. At stage n, the spanning tree T isapproximated by a finite tree Tn rooted at the identity.The approximation Tn+1 is obtained by connecting edges to the points of V that are not already vertices of Tn but can be obtained from vertices of Tn via multiplication by a random walk step taking values in the generating set of V. This construction leads to a compactification of the semigroup V inwhich a sequence of elements of V that is not eventually constant is convergent if the random geodesic through the spanning tree T that joins the identity to the nth element of the sequence converges in distribution as n→∞. The compactification is identified in a number of examples. Also, it is shown that if h(Tn) and #(Tn) denote, respectively, the height and size of the approximating tree Tn, then there are constants 0<ch≤1 and 0≥c# ≤log2 such that limn→∞ n–1 h(Tn)= ch and limn→∞n–1 log# (Tn)= c# almost surely.

scholarly journals Cayley Graphs on the Symmetric Group Generated by Initial Reversals have Unit Spectral Gap

10.37236/267 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Filippo Cesi
Keyword(s):  
Irreducible Representation ◽  
Symmetric Group ◽  
Empirical Evidence ◽  
Cayley Graph ◽  
Simple Proof ◽  
Spectral Gap ◽  
Cayley Graphs ◽  
Complete Spectrum ◽  
Nonzero Eigenvalue ◽  
Direct Sum

In a recent paper Gunnells, Scott and Walden have determined the complete spectrum of the Schreier graph on the symmetric group corresponding to the Young subgroup $S_{n-2}\times S_2$ and generated by initial reversals. In particular they find that the first nonzero eigenvalue, or spectral gap, of the Laplacian is always 1, and report that "empirical evidence" suggests that this also holds for the corresponding Cayley graph. We provide a simple proof of this last assertion, based on the decomposition of the Laplacian of Cayley graphs, into a direct sum of irreducible representation matrices of the symmetric group.

scholarly journals Dominating sets in Cayley graphs on $ Z_{n} $

2007 ◽  
Vol 38 (4) ◽  
pp. 341-345 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
I. Rani
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Generating Set ◽  
Minimal Dominating Set

A Cayley graph is a graph constructed out of a group $ \Gamma $ and its generating set $ A $. In this paper we attempt to find dominating sets in Cayley graphs constructed out of $ Z_{n} $. Actually we find the value of domination number for $ Cay(Z_{n}, A) $ and a minimal dominating set when $ |A| $ is even and further we have proved that $ Cay(Z_{n}, A) $ is excellent. We have also shown that $ Cay(Z_{n}, A) $ is $ 2- $excellent, when $ n = t(|A|+1)+1 $ for some integer $ t, t>0 $.

scholarly journals Cayley Graphs of Order 27p Are Hamiltonian

2011 ◽  
Vol 2011 ◽  
pp. 1-16 ◽  
Author(s):  
Ebrahim Ghaderpour ◽  
Dave Witte Morris
Keyword(s):  
Finite Group ◽  
Cayley Graph ◽  
Hamiltonian Cycle ◽  
Cayley Graphs ◽  
Generating Set ◽  
A Finite Group

Suppose that G is a finite group, such that |G|=27p, where p is prime. We show that if S is any generating set of G, then there is a Hamiltonian cycle in the corresponding Cayley graph Cay (G;S).

scholarly journals Differential operators and Higher Specht polynomials

2021 ◽  
Vol 2090 (1) ◽  
pp. 012096
Author(s):  
Ibrahim Nonkané ◽  
Léonard Todjihounde
Keyword(s):  
Symmetric Group ◽  
Polynomial Ring ◽  
Weyl Algebra ◽  
Differential Operators ◽  
Symmetric Groups ◽  
Symmetric Polynomial ◽  
Symmetric Polynomials ◽  
Polynomial Representation ◽  
Invariant Differential ◽  
Moser System

Abstract In this note, we study the action of the rational quantum Calogero-Moser system on polynomials rings. This a continuation of our paper [Ibrahim Nonkan 2021 J. Phys.: Conf. Ser. 1730 012129] in which we deal with the polynomial representation of the ring of invariant differential operators. Using the higher Specht polynomials we give a detailed description of the actions of the Weyl algebra associated with the ring of the symmetric polynomial C[x 1,..., xn]Sn on the polynomial ring C[x 1,..., xn ]. In fact, we show that its irreducible submodules over the ring of differential operators invariant under the symmetric group are its submodules generated by higher Specht polynomials over the ring of the symmetric polynomials. We end up studying the polynomial representation of the ring of differential operators invariant under the actions of products of symmetric groups by giving the generators of its simple components, thus we give a differential structure to the higher Specht polynomials.

scholarly journals Automorphisms and isomorphisms of enhanced hypercubes

Filomat ◽  
2020 ◽  
Vol 34 (8) ◽  
pp. 2805-2812
Author(s):  
Lu Lu ◽  
Qiongxiang Huang
Keyword(s):  
Automorphism Group ◽  
Vector Space ◽  
Cayley Graph ◽  
Cayley Graphs ◽  
Standard Basis ◽  
Generating Set ◽  
Folded Hypercube

Let Zn2 be the elementary abelian 2-group, which can be viewed as the vector space of dimension n over F2. Let {e1,..., en} be the standard basis of Zn2 and ?k = ek +...+ en for some 1 ? k ? n-1. Denote by ?n,k the Cayley graph over Zn2 with generating set Sk = {e1,..., en,?k}, that is, ?n,k = Cay(Zn2,Sk). In this paper, we characterize the automorphism group of ?n,k for 1 ? k ? n-1 and determine all Cayley graphs over Zn2 isomorphic to ?n,k. Furthermore, we prove that for any Cayley graph ? = Cay(Zn2,T), if ? and ?n,k share the same spectrum, then ? ? ?n,k. Note that ?n,1 is known as the so called n-dimensional folded hypercube FQn, and ?n,k is known as the n-dimensional enhanced hypercube Qn,k.

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