On Cayley Graphs on the Symmetric Group Generated by Tranpositions

COMBINATORICA ◽  
2000 ◽  
Vol 20 (4) ◽  
pp. 505-519 ◽  
Author(s):  
Joel Friedman
Keyword(s):  
Symmetric Group ◽  
Cayley Graphs

Bounded Dilation Maps of Hypercubes into Cayley Graphs on the Symmetric Group

1996 ◽  
Vol 29 (6) ◽  
pp. 551 ◽  
Author(s):  
Z. Miller ◽  
D. Pritikin ◽  
I.H. Sudborough
Keyword(s):  
Symmetric Group ◽  
Cayley Graphs

scholarly journals On the diameter of cayley graphs of the symmetric group

1988 ◽  
Vol 49 (1) ◽  
pp. 175-179 ◽  
Author(s):  
László Babai ◽  
Ákos Seress
Keyword(s):  
Symmetric Group ◽  
Cayley Graphs

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.

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}.

Bounded dilation maps of hypercubes into Cayley graphs on the symmetric group

1996 ◽  
Vol 29 (6) ◽  
pp. 551-572 ◽  
Author(s):  
Z. Miller ◽  
D. Pritikin ◽  
I. H. Sudborough
Keyword(s):  
Symmetric Group ◽  
Cayley Graphs

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.

scholarly journals Bounds on the diameter of Cayley graphs of the symmetric group

2013 ◽  
Vol 40 (1) ◽  
pp. 1-22 ◽  
Author(s):  
John Bamberg ◽  
Nick Gill ◽  
Thomas P. Hayes ◽  
Harald A. Helfgott ◽  
Ákos Seress ◽  
...  
Keyword(s):  
Symmetric Group ◽  
Cayley Graphs

scholarly journals Spectrum of Cayley graphs on the symmetric group generated by transpositions

2012 ◽  
Vol 437 (3) ◽  
pp. 1033-1039 ◽  
Author(s):  
Roi Krakovski ◽  
Bojan Mohar
Keyword(s):  
Symmetric Group ◽  
Cayley Graphs

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.

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