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}.

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 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 Random Cayley Graphs are Expanders: a Simple Proof of the Alon–Roichman Theorem

10.37236/1815 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Zeph Landau ◽  
Alexander Russell
Keyword(s):  
Finite Group ◽  
Cayley Graph ◽  
Group Algebra ◽  
Simple Proof ◽  
Cayley Graphs ◽  
Random Variables ◽  
Irreducible Representations ◽  
Original Proof ◽  
A Finite Group ◽  
Second Eigenvalue

We give a simple proof of the Alon–Roichman theorem, which asserts that the Cayley graph obtained by selecting $c_\varepsilon \log |G|$ elements, independently and uniformly at random, from a finite group $G$ has expected second eigenvalue no more than $\varepsilon$; here $c_\varepsilon$ is a constant that depends only on $\varepsilon$. In particular, such a graph is an expander with constant probability. Our new proof has three advantages over the original proof: (i.) it is extremely simple, relying only on the decomposition of the group algebra and tail bounds for operator-valued random variables, (ii.) it shows that the $\log |G|$ term may be replaced with $\log D$, where $D \leq |G|$ is the sum of the dimensions of the irreducible representations of $G$, and (iii.) it establishes the result above with a smaller constant $c_\varepsilon$.

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

On the Diameter of Random Cayley Graphs of the Symmetric Group

1992 ◽  
Vol 1 (3) ◽  
pp. 201-208 ◽  
Author(s):  
L. Babai ◽  
G. L. Hetyei
Keyword(s):  
Symmetric Group ◽  
Uniform Distribution ◽  
Cayley Graph ◽  
Cayley Graphs ◽  
Alternating Group

Let σ, π be two permutations selected at random from the uniform distribution on the symmetric group Sn. By a result of Dixon [5], the subgroup G generated by σ, π is almost always (i.e. with probability approaching 1 as n → ∞) either Sn or the alternating group An. We prove that the diameter of the Cayley graph of G defined by {σ, π} is almost always not greater than exp ((½ + o(l)). (In n)2).

scholarly journals Cayley Graphs Without a Bounded Eigenbasis

2020 ◽  
Author(s):  
Ashwin Sah ◽  
Mehtaab Sawhney ◽  
Yufei Zhao
Keyword(s):  
Cayley Graph ◽  
Orthonormal Basis ◽  
Cayley Graphs ◽  
The Other ◽  
Transitive Graph ◽  
Classic Result ◽  
Other Hand ◽  
Small Set ◽  
Vertex Transitive ◽  
Vertex Transitive Graph

Abstract Does every $n$-vertex Cayley graph have an orthonormal eigenbasis all of whose coordinates are $O(1/\sqrt{n})$? While the answer is yes for abelian groups, we show that it is no in general. On the other hand, we show that every $n$-vertex Cayley graph (and more generally, vertex-transitive graph) has an orthonormal basis whose coordinates are all $O(\sqrt{\log n / n})$, and that this bound is nearly best possible. Our investigation is motivated by a question of Assaf Naor, who proved that random abelian Cayley graphs are small-set expanders, extending a classic result of Alon–Roichman. His proof relies on the existence of a bounded eigenbasis for abelian Cayley graphs, which we now know cannot hold for general groups. On the other hand, we navigate around this obstruction and extend Naor’s result to nonabelian groups.

scholarly journals Normal Edge-Transitive Cayley Graphs of the Group U6n

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
A. Assari ◽  
F. Sheikhmiri
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
The Right ◽  
Edge Transitive

A Cayley graph of a group G is called normal edge-transitive if the normalizer of the right representation of the group in the automorphism of the Cayley graph acts transitively on the set of edges of the graph. In this paper, we determine all connected normal edge-transitive Cayley graphs of the group U6n.

scholarly journals Approximating Cayley Diagrams Versus Cayley Graphs

2012 ◽  
Vol 21 (4) ◽  
pp. 635-641
Author(s):  
ÁDÁM TIMÁR
Keyword(s):  
Cayley Graph ◽  
Hamiltonian Cycle ◽  
Cayley Graphs ◽  
The Other ◽  
Finite Graphs ◽  
Similar Construction ◽  
Graph Property

We construct a sequence of finite graphs that weakly converge to a Cayley graph, but there is no labelling of the edges that would converge to the corresponding Cayley diagram. A similar construction is used to give graph sequences that converge to the same limit, and such that a Hamiltonian cycle in one of them has a limit that is not approximable by any subgraph of the other. We give an example where this holds, but convergence is meant in a stronger sense. This is related to whether having a Hamiltonian cycle is a testable graph property.

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