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

Huang, Xueyi; Huang, Qiongxiang; Cioabă, Sebastian M.

doi:10.37236/8054

Cited by 8 publications

(13 citation statements)

References 34 publications

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“…The following lemma is straightforward and is given without a proof (see [8,Lemma 5]). Let G := Sym(n) and for any i ∈ {1, 2, .…”

Section: 2mentioning

confidence: 99%

“…, n}. If B Π i = (b s,t ) s,t∈{1,2,...,n} is the quotient matrix corresponding to the equitable partition Π i of Γ n,k given by G/G i , then by [8] b…”

Section: 2mentioning

confidence: 99%

“…This eigenvalue corresponds to the irreducible character of the partition [n − 1, 1]. Moreover, Huang et al [8] proved that λ 2 (Γ n,n−4 ) = n(n−2)(n−3)(n−5)…”

mentioning

confidence: 99%

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On the second eigenvalue of a Cayley graph of the symmetric group

Roghayeh¹,

Sarobidy²

2021

Preprint

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In 2020, Siemons and Zalesski [On the second eigenvalue of some Cayley graphs of the symmetric group. arXiv preprint arXiv:2012.12460, 2020] determined the second eigenvalue of the Cayley graph Γ n,k = Cay(Sym(n), C(n, k)) for k = 0 and k = 1, where C(n, k) is the conjugacy class of (n − k)-cycles. In this paper, it is proved that for any n ≥ 3 and k ∈ N relatively small compared to n, the second eigenvalue of Γ n,k is the eigenvalue afforded by the irreducible character of Sym(n) that corresponds to the partition [n − 1, 1]. As a byproduct of our method, the result of Siemons and Zalesski when k ∈ {0, 1} is retrieved. Moreover, we prove that the second eigenvalue of Γn,n−5 is also equal to the eigenvalue afforded by the irreducible character of the partition [n − 1, 1].

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“…The following lemma is straightforward and is given without a proof (see [8,Lemma 5]). Let G := Sym(n) and for any i ∈ {1, 2, .…”

Section: 2mentioning

confidence: 99%

“…, n}. If B Π i = (b s,t ) s,t∈{1,2,...,n} is the quotient matrix corresponding to the equitable partition Π i of Γ n,k given by G/G i , then by [8] b…”

Section: 2mentioning

confidence: 99%

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On the second eigenvalue of a Cayley graph of the symmetric group

Roghayeh¹,

Sarobidy²

2021

Preprint

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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. One of the eigenvalues in Theorem 5.2 can also be determined by[7, Lemma 11].…”

mentioning

confidence: 99%

“…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. One of the eigenvalues in Theorem 5.2 can also be determined by[7, Lemma 11].…”

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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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On the Second Eigenvalue of Certain Cayley Graphs on the Symmetric Group

Roghayeh

Sarobidy

2023

Bull. Malays. Math. Sci. Soc.

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

Cited by 8 publications

References 34 publications

On the second eigenvalue of a Cayley graph of the symmetric group

On the second eigenvalue of a Cayley graph of the symmetric group

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

On the Second Eigenvalue of Certain Cayley Graphs on the Symmetric Group

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