Aldous’s spectral gap conjecture for normal sets

Parzanchevski, Ori; Puder, Doron

doi:10.1090/tran/8155

Cited by 7 publications

(13 citation statements)

References 26 publications

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“…We can also define whether a weighted Cayley graph on any finite group has the Aldous property with respect to a representation of the group. However, since this requires an equivalent version of Aldous' spectral gap conjecture which would be a significant deviation from the goal of this paper, we refer the reader to [7,24,32] for this definition. In [4], Cesi proved that the pancake graph P n has the Aldous property.…”

Section: Introductionmentioning

confidence: 99%

“…In [4], Cesi proved that the pancake graph P n has the Aldous property. In [32], Parzanchevski and Puder studied the strictly second largest eigenvalue of Cay(S n , S) in the case when S is a single conjugacy class of S n . In [19], Huang, Huang and Cioabȃ proved that a majority of the connected normal Cayley graphs on S n (n ≥ 7) with connection sets consisting of permutations moving at most five points possess the Aldous property.…”

Section: Introductionmentioning

confidence: 99%

“…In [32,Proposition 2.4], it was proved that, for sufficiently large n and any conjugacy class S of S n whose elements fix at most one point, the strictly second largest eigenvalue of Cay(S n , S) is attained by one of the following eight irreducible representations of S n :…”

Section: Introductionmentioning

confidence: 99%

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Aldous' spectral gap property for normal Cayley graphs on symmetric groups

Li¹,

Xia²,

Zhou³

2022

Preprint

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Aldous' spectral gap conjecture states that the second largest eigenvalue of any connected Cayley graph on the symmetric group S n with respect to a set of transpositions is achieved by the standard representation of S n . This celebrated conjecture, which was proved in its general form in 2010, has inspired much interest in searching for other families of Cayley graphs on S n with the property that the largest eigenvalue strictly smaller than the degree is attained by the standard representation of S n . In this paper, we prove three results on normal Cayley graphs on S n possessing this property for sufficiently large n, one of which can be viewed as a generalization of the "normal" case of Aldous' spectral gap conjecture.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Aldous' spectral gap property for normal Cayley graphs on symmetric groups

Li¹,

Xia²,

Zhou³

2022

Preprint

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“…In [12], Chung and Tobin determined the second eigenvalues of the reversal graph R n = Cay(S n , {r i,j | 1 ≤ i < j ≤ n}) and a family of graphs that generalize the pancake graph P n . In [32], Parzanchevski and Puder proved that, for large enough n, if S ⊆ S n is a full conjugacy class generating S n then the second eigenvalue of Cay(S n , S) is always associated with one of eight low-dimensional representations of S n . In [25], the authors determined the second eigenvalues of the alternating group graph AG n = Cay(A n , {(1, 2, i), (1, i, 2) | 3 ≤ i ≤ n}) (introduced by Jwo, Lakshmivarahan and Dhall [27]), the extended alternating group graph EAG n = Cay(A n , {(1, i, j), (1, j, i) | 2 ≤ i < j ≤ n}) and the complete alternating group graph CAG n = Cay(A n , {(i, j, k), (i, k, j) | 1 ≤ i < j < k ≤ n}) (defined by Huang and Huang [24]).…”

Section: Introductionmentioning

confidence: 99%

The Second Eigenvalue of some Normal Cayley Graphs of Highly Transitive Groups

Huang

Cioabă

2019

Electron. J. Combin.

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Let Γ be a finite group acting transitively on [n] = {1, 2, . . . , n}, and let G = Cay(Γ, T ) be a Cayley graph of Γ. The graph G is called normal if T is closed under conjugation. In this paper, we obtain an upper bound for the second (largest) eigenvalue of the adjacency matrix of the graph G in terms of the second eigenvalues of certain subgraphs of G (see Theorem 2.6). Using this result, we develop a recursive method to determine the second eigenvalues of certain Cayley graphs of S n and we determine the second eigenvalues of a majority of the connected normal Cayley graphs (and some of their subgraphs) of S n with max τ ∈T |supp(τ )| ≤ 5, where supp(τ ) is the set of points in [n] non-fixed by τ .

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“…Final NoteWe would like to mention the recent asymptotic results by O. Parzanchevski & Puder[15] and R. Maleki & A. S. Razafimahatratra[14] relating to Theorems 1.1 and 1.2. Furthermore, recently X. Huang, Q. Huang & S. M. Cioabȃ[7] have computed the second largest eigenvalue of some Cayley graphs for highly transitive permutation groups on the natural permutation module.…”

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confidence: 99%

On the second largest eigenvalue of some Cayley graphs of the symmetric group

Siemons

Zalesski

2021

J Algebr Comb

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Let $$S_n$$ S n and $$A_{n}$$ A n denote the symmetric and alternating group on the set $$\{1,\ldots ,n\},$$ { 1 , … , n } , respectively. In this paper we are interested in the second largest eigenvalue $$\lambda _{2}(\Gamma )$$ λ 2 ( Γ ) of the Cayley graph $$\Gamma =\mathrm{Cay}(G,H)$$ Γ = Cay ( G , H ) over $$G=S_{n}$$ G = S n or $$A_{n}$$ A n for certain connecting sets H. Let $$1<k\le n$$ 1 < k ≤ n and denote the set of all k-cycles in $$S_{n}$$ S n by C(n, k). For $$H=C(n,n)$$ H = C ( n , n ) we prove that $$\lambda _{2}(\Gamma )=(n-2)!$$ λ 2 ( Γ ) = ( n - 2 ) ! (when n is even) and $$\lambda _{2}(\Gamma )=2(n-3)!$$ λ 2 ( Γ ) = 2 ( n - 3 ) ! (when n is odd). Further, for $$H=C(n,n-1)$$ H = C ( n , n - 1 ) we have $$\lambda _{2}(\Gamma )=3(n-3)(n-5)!$$ λ 2 ( Γ ) = 3 ( n - 3 ) ( n - 5 ) ! (when n is even) and $$\lambda _{2}(\Gamma )=2(n-2)(n-5) !$$ λ 2 ( Γ ) = 2 ( n - 2 ) ( n - 5 ) ! (when n is odd). The case $$H=C(n,3)$$ H = C ( n , 3 ) has been considered in Huang and Huang (J Algebraic Combin 50:99–111, 2019). Let $$1\le r<k<n$$ 1 ≤ r < k < n and let $$C(n,k;r) \subseteq C(n,k)$$ C ( n , k ; r ) ⊆ C ( n , k ) be set of all k-cycles in $$S_{n}$$ S n which move all the points in the set $$\{1,2,\ldots ,r\}.$$ { 1 , 2 , … , r } . That is to say, $$g=(i_{1},i_{2},\ldots ,i_{k})(i_{k+1})\dots (i_{n})\in C(n,k;r)$$ g = ( i 1 , i 2 , … , i k ) ( i k + 1 ) ⋯ ( i n ) ∈ C ( n , k ; r ) if and only if $$\{1,2,\ldots ,r\}\subset \{i_{1},i_{2},\ldots ,i_{k}\}.$$ { 1 , 2 , … , r } ⊂ { i 1 , i 2 , … , i k } . Our main result concerns $$\lambda _{2}(\Gamma )$$ λ 2 ( Γ ) , where $$\Gamma =\mathrm{Cay}(G,H)$$ Γ = Cay ( G , H ) with $$H=C(n,k;r)$$ H = C ( n , k ; r ) with $$1\le r<k<n$$ 1 ≤ r < k < n when $$G=S_{n}$$ G = S n if k is even and $$G=A_{n}$$ G = A n if k is odd. Here we observe that $$\begin{aligned} \lambda _{2}(\Gamma )\ge (k-2)! {n-r \atopwithdelims ()k-r} \frac{1}{n-r} \big ((k-1)(n-k) - \frac{(k-r-1)(k-r)}{n-r-1}\big ). \end{aligned}$$ λ 2 ( Γ ) ≥ ( k - 2 ) ! n - r k - r 1 n - r ( ( k - 1 ) ( n - k ) - ( k - r - 1 ) ( k - r ) n - r - 1 ) . We prove that this bound is attained in the special case $$k=r+1$$ k = r + 1 , giving $$\lambda _{2}(\Gamma )=r!(n-r-1)$$ λ 2 ( Γ ) = r ! ( n - r - 1 ) . The cases with $$H=C(n,3;1)$$ H = C ( n , 3 ; 1 ) and $$H=C(n,3;2)$$ H = C ( n , 3 ; 2 ) were considered earlier in [6].

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Aldous’s spectral gap conjecture for normal sets

Cited by 7 publications

References 26 publications

Aldous' spectral gap property for normal Cayley graphs on symmetric groups

Aldous' spectral gap property for normal Cayley graphs on symmetric groups

The Second Eigenvalue of some Normal Cayley Graphs of Highly Transitive Groups

On the second largest eigenvalue of some Cayley graphs of the symmetric group

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