The smallest eigenvalues of the 1-point fixing graph

Linear Algebra and its Applications 493 (2016) 433–446

Contents lists available at ScienceDirect

Linear Algebra and its Applications www.elsevier.com/locate/laa

The smallest eigenvalues of the 1-point fixing graph Cheng Yeaw Ku a , Terry Lau b , Kok Bin Wong b,∗ a

Department of Mathematics, National University of Singapore, Singapore 117543, Singapore b Institute of Mathematical Sciences, University of Malaya, 50603 Kuala Lumpur, Malaysia

a r t i c l e

i n f o

Article history: Received 12 March 2015 Accepted 10 December 2015 Available online xxxx Submitted by R. Brualdi MSC: 05C50 05A05

a b s t r a c t Let Sn be the symmetric group on {1, . . . , n}. The k-point fixing graph F (n, k) is defined to be the graph with vertex set Sn and two vertices g, h of F (n, k) are joined if and only if gh−1 fixes exactly k points. In this paper, we determine the smallest eigenvalue for F (n, 1). © 2015 Elsevier Inc. All rights reserved.

Keywords: Arrangement graph Cayley graphs Symmetric group

1. Introduction Let G be a finite group and S be an inverse closed subset of G, i.e., 1 ∈ / S and s ∈ S ⇒ s−1 ∈ S. The Cayley graph Γ(G, S) is the graph which has the elements of G as its vertices and two vertices u, v ∈ G are joined by an edge if and only if v = su for some s ∈ S. * Corresponding author. E-mail addresses: [email protected] (C.Y. Ku), [email protected] (T. Lau), [email protected] (K.B. Wong). http://dx.doi.org/10.1016/j.laa.2015.12.011 0024-3795/© 2015 Elsevier Inc. All rights reserved.

434

C.Y. Ku et al. / Linear Algebra and its Applications 493 (2016) 433–446

A Cayley graph Γ(G, S) is said to be normal if S is closed under conjugation. It is well known that the eigenvalues of a normal Cayley graph Γ(G, S) can be expressed in terms of the irreducible characters of G. Theorem 1.1. (See [1,5,18,19].) The eigenvalues of a normal Cayley graph Γ(G, S) are given by ηχ =

1  χ(s), χ(1) s∈S

where χ ranges over all the irreducible characters of G. Moreover, the multiplicity of ηχ is χ(1)2 . Let Sn be the symmetric group on [n] = {1, . . . , n} and S ⊆ Sn be closed under conjugation. Since central characters are algebraic integers [12, Theorem 3.7 on p. 36] and that the characters of the symmetric group are integers ([12, 2.12 on p. 31] or [22, Corollary 2 on p. 103]), by Theorem 1.1, the eigenvalues of Γ(Sn , S) are integers. Corollary 1.2. The eigenvalues of a normal Cayley graph Γ(Sn , S) are integers. For k ≤ n, a k-permutation of [n] is an injective function from [k] to [n]. So any k-permutation π can be represented by a vector (i1 , . . . , ik ) where π(j) = ij for j = 1, . . . , k. Let 1 ≤ r ≤ k ≤ n. The (n, k, r)-arrangement graph A(n, k, r) has all the k-permutations of [n] as vertices and two k-permutations are adjacent if they differ in exactly r positions. Formally, the vertex set V (n, k) and edge set E(n, k, r) of A(n, k, r) are   V (n, k) = (p1 , p2 , . . . , pk ) | pi ∈ [n] and pi = pj for i = j ,  E(n, k, r) = {(p1 , p2 , . . . , pk ), (q1 , q2 , . . . , qk )} | pi = qi for i ∈ R and

 pj = qj for all j ∈ [k] \ R for some R ⊆ [k] with |R| = r .

Note that |V (n, k)| = n!/(n − k)! and A(n, k, r) is a regular graph [3, Theorem 4.2]. In particular, A(n, k, 1) is a k(n − k)-regular graph. We note here that A(n, k, 1) was first introduced in [4] as an interconnection network model for parallel computation. Furthermore, A(n, k, 1) is called the partial permutation graph by Krakovski and Mohar in [13]. The eigenvalues of the arrangement graphs A(n, k, 1) were first studied in [2] by using a method developed by Godsil and McKay [10]. A relation between the eigenvalues of A(n, k, r) and certain Cayley graphs was given in [3]. The derangement graph Γn is the Cayley graph Γ(Sn , Dn ) where Dn is the set of derangements in Sn . That is, two vertices g, h of Γn are joined if and only if g(i) = h(i) for all i ∈ [n], or equivalently gh−1 fixes no point. Since Dn is closed under conjugation, by Corollary 1.2, the eigenvalues of the derangement graph are integers. The lower and

C.Y. Ku et al. / Linear Algebra and its Applications 493 (2016) 433–446

435

upper bounds of the absolute values of these eigenvalues have been studied in [16,17,20]. Note that derangement graph is a kind of arrangement graph, i.e., Γn = A(n, n, n). Let 0 ≤ k < n and S(n, k) be the set of all σ ∈ Sn such that σ fixes exactly k elements. Note that S(n, k) is an inverse closed subset of Sn . The k-point fixing graph is defined to be F(n, k) = Γ(Sn , S(n, k)). That is, two vertices g, h of F(n, k) are joined if and only if gh−1 fixes exactly k points. Note that the k-point fixing graph is also a kind of arrangement graph, i.e., F(n, k) = A(n, n, n − k). Furthermore, the 0-point fixing graph is the derangement graph, i.e., F(n, 0) = Γn = A(n, n, n). Clearly, F(n, k) is vertex-transitive, so it is |S(n, k)|-regular and the largest eigenvalue of F(n, k) is |S(n, k)|. Furthermore, S(n, k) is closed under conjugation. Therefore, by Corollary 1.2, the eigenvalues of the k-point fixing graph are integers. Eigenvalues of F(n, k) were first studied in [14]. Recall that a partition λ of n, denoted by λ  n, is a weakly decreasing sequence λ1 ≥ . . . ≥ λr with λr ≥ 1 such that λ1 + · · · + λr = n. We write λ = (λ1 , . . . , λr ). The size of λ, denoted by |λ|, is n and each λi is called the i-th part of the partition. We also use the notation (μa1 1 , . . . , μas s )  n to denote the partition where μi are the distinct nonzero parts that occur with multiplicity ai . For example, (5, 5, 4, 4, 2, 2, 2, 1) ←→ (52 , 42 , 23 , 1). It is well known that both the conjugacy classes of Sn and the irreducible characters of Sn are indexed by partitions λ of [n]. Since S(n, k) is closed under conjugation, the eigenvalue ηχλ (k) of the k-point fixing graph can be denoted by ηλ (k). Throughout the paper, we shall use this notation. Renteln [20, Theorem 7.1] showed that the smallest eigenvalue of the derangement graph F(n, 0) is η(n−1,1) (0) = −

|Dn | . n−1

A generalization of his result was given in [15]. In this paper, we will prove the following theorem. Theorem 1.3. For n ≥ 7, the smallest eigenvalue of F(n, 1) is equal to η(n−2,2) (1) = −

1 (dn−1 + (−1)n (n − 2)) (n − 3)

1 where dn = |Dn |. Furthermore, ηλ (1) = − (n−3) (dn−1 + (−1)n (n − 2)) if and only if λ = (n − 2, 2).

436

C.Y. Ku et al. / Linear Algebra and its Applications 493 (2016) 433–446

The paper is organized as follows. In Section 2, we will provide some known results regarding the eigenvalues of F(n, 0). In Section 3, we will give the exact values of the eigenvalues ηλ (1) for λ ∈ {(n), (1n ), (n − 1, 1), (2, 1n−2 ), (n − 2, 2), (22 , 1n−4 ), (n − 2, 12 ), (3, 1n−3 )} and give an upper bound of |ηλ (1)| for λ ∈ / {(n), (1n ), (n − 1, 1), (2, 1n−2 ), (n − 2, 2), (22 , 1n−4 ), (n − 2, 12 ), (3, 1n−3 )}. In Section 4, we will prove Theorem 1.3. 2. Eigenvalues of F (n, 0) Renteln [20] gave a recurrence formula for the eigenvalues of F(n, 0). To describe this result, we require some terminology. To the Ferrers diagram of a partition λ, we assign xy-coordinates to each of its boxes by defining the upper-left-most box to be (1, 1), with the x axis increasing to the right and the y axis increasing downwards. Then the hook of λ is the union of the boxes (x , 1) and (1, y  ) of the Ferrers diagram of λ, where x ≥ 1, y  ≥ 1. Let  cλ hλ denote the hook of λ and let hλ denote the size of  hλ . Similarly, let  and cλ denote the first column of λ and the size of  cλ respectively. Note that cλ is equal to the number of rows of λ. When λ is clear from the context, we will replace  cλ hλ , hλ ,    and cλ by h, h,  c and c respectively. Let λ − h  n − h denote the partition obtained from λ by removing its hook. Also, let λ −  c denote the partition obtained from λ by removing the first column of its Ferrers diagram, i.e. (λ1 , . . . , λr ) −  c = (λ1 − 1, . . . , λr − 1)  n − r. Theorem 2.1 (Renteln’s recurrence formula). (See [20, Theorem 6.5].) For any partition λ = (λ1 , . . . , λr )  n, the eigenvalues of the derangement graph F(n, 0) satisfy the following recurrence: ηλ (0) = (−1)h ηλ−h (0) + (−1)h+λ1 hηλ−c (0) with initial condition η∅ (0) = 1. Later, Ku and Wong [17] gave another recurrence formula for the eigenvalues of F(n, 0). To describe this result, for a partition λ = (λ1 , . . . , λr )  n, let  lλ denote the last row of λ and let lλ denote the size of  lλ lλ . Clearly, we have lλ = λr . Let λ −  denote the partition obtained from λ by deleting the last row. When λ is clear from the context, we will replace  lλ , lλ by  l and l respectively. Theorem 2.2 (Ku–Wong’s recurrence formula). (See [17, Theorem 1.4].) For any partition λ = (λ1 , . . . , λr )  n, the eigenvalues of the derangement graph F(n, 0) satisfy the following recurrence: ηλ (0) = (−1)λr ηλ−l (0) + (−1)r−1 λr ηλ−c (0) with initial condition η∅ (0) = 1.

C.Y. Ku et al. / Linear Algebra and its Applications 493 (2016) 433–446

437

3. Eigenvalues of F (n, 1) The terms and notations used here are standard in the representation theory of symmetric groups. For undefined terms, we refer the reader to Stanley [23] or Fulton and Harris [9]. The argument used here is similar to that of Ellis [6]. So, for the undefined terms, the reader may also refer to [6, Sections 2.1, 2.2]. Let λ  n. For a box with coordinate (a, b) in the Ferrers diagram of λ, the hook-length hλ (a, b) is the size of the set of all the boxes with coordinate (i, j) where i = a and j ≥ b, or i ≥ a and j = b. A well-known theorem of Frame, Robinson and Thrall states that if λ is a partition of n, then the number f λ of Standard Young Tableaux of shape λ is n! divided by the product of the hook-lengths [8]. Lemma 3.1 (Hook-length formula). f λ = χλ (1) = 

n! , hλ (a, b)

where the product is over all the boxes (a, b) in the Ferrers diagram of λ. The hook-length formula is of great importance in the representation theory of the symmetric group. Indeed, the Specht modules S λ , indexed by partitions λ of n, form a complete set of irreducible representations of Sn , where the dimension of the Specht module S λ is known to be equal to the number f λ . Lemma 3.2. (See [6, Lemma 2.4].) For n ≥ 9, the only Specht modules S λ of dimension   f λ < n−1 − 1 are as follows: 2 (a) (b) (c) (d)

S (n) (the trivial representation), dimension 1; n S (1 ) (the sign representation), dimension 1; S (n−1,1) , dimension n − 1; n−2 n S (2,1 ) (∼ = S (1 ) ⊗ S (n−1,1) ), dimension n − 1.

The Branching Theorem (see [21, Theorem 2.8.3]) states that for any partition λ of n, the restriction [λ] ↓ Sn−1 is isomorphic to a direct sum of those irreducible representations [μ] of Sn−1 such that the Young diagram of μ can be obtained from that of λ by deleting a single dot, i.e., if λi− is the partition whose Young diagram is obtained by deleting the dot at the end of the i-th row of that of λ, then [λ] ↓ Sn−1 =



[λi− ].

(1)

i:λi >λi−1

Lemma 3.3. For n ≥ 13, the only Specht modules S λ of dimension 1 6 n(n − 1)(n − 5) are as follows:

n−1 2

− 1 ≤ fλ <

438

C.Y. Ku et al. / Linear Algebra and its Applications 493 (2016) 433–446

(a) S (n−2,2) , dimension

n−1

− 1; 2n−1 (b) S , dimension 2 − 1;   (n−2,12 ) , dimension n−1 (c) S 2 ;   n−3 (d) S (3,1 ) , dimension n−1 2 . (22 ,1n−4 )

Proof. By Lemma 3.2, it is sufficient to prove the following statement: (∗) For n ≥ 13, f λ < 16 n(n − 1)(n − 5) if and only if   λ ∈ (n), (n − 1, 1), (1n ), (2, 1n−2 ), (n − 2, 2), (22 , 1n−4 ), (n − 2, 12 ), (3, 1n−3 ) . By a direct calculation using Lemma 3.1, (∗) can be verified for n = 13, 14. We proceed by induction. Assume that (∗) holds for n − 2, n − 1; we will prove it for n. Let α be a partition of n such that f α < 16 n(n − 1)(n − 5). Consider the restriction [α] ↓ Sn−1 , which has the same dimension. First suppose [α] ↓ Sn−1 is reducible. If it has one of the eight irreducible representations (in (∗)) as a constituent, then by (1), the possibilities of α are as follows: Constituent

Possibilities of α

[n − 1] [1n−1 ] [n − 2, 1] [2, 1n−3 ] [n − 3, 2] [22 , 1n−5 ] [n − 3, 12 ] [3, 1n−4 ]

(n), (n − 1, 1) (1n ), (2, 1n−1 ) (n − 1, 1), (n − 2, 2), (n − 2, 12 ) (2, 1n−2 ), (22 , 1n−4 ), (3, 1n−3 ) (n − 2, 2), (n − 3, 3), (n − 3, 2, 1) (3, 2, 1n−5 ), (23 , 1n−6 ), (22 , 1n−4 ) (n − 2, 12 ), (n − 3, 2, 1), (n − 3, 13 ) (4, 1n−4 ), (3, 2, 1n−5 ), (3, 1n−3 )

By Lemma 3.1, the new irreducible representations above all have dimension ≥ 16 n(n − 1)(n − 5): fα

α (n − 3, 3), (2 , 1 ) (n − 3, 2, 1), (3, 2, 1n−5 ) (n − 3, 13 ), (4, 1n−4 ) 3

n−6

1 6 n(n − 1)(n − 5) 1 3 n(n − 2)(n − 4) 1 6 (n − 1)(n − 2)(n

− 3)

Hence, (∗) holds, provided that [α] ↓ Sn−1 has one of the eight irreducible representations in (∗) as a constituent. Suppose that the irreducible constituents of [α] ↓ Sn−1 do not include any of the eight irreducible representations in (∗). By induction hypothesis for n −1, each irreducible con  stituent has dimension ≥ 16 (n − 1)(n − 2)(n − 6). Note that 2 16 (n − 1)(n − 2)(n − 6) > 1 6 n(n − 1)(n − 5) for n ≥ 15. Thus, [α] ↓ Sn−1 has exactly one irreducible constituent, i.e., it is irreducible. Therefore, [α] = [st ] for some s, t ∈ N with st = n, i.e., it has a square Young diagram. Furthermore, t ≥ 2.

C.Y. Ku et al. / Linear Algebra and its Applications 493 (2016) 433–446

439

Now consider [α] ↓ Sn−2 = [st−1 , s − 2] + [st−2 , s − 1, s − 1]. Note that for n ≥ 15, neither of these two irreducible constituents are any of the eight irreducible representations in (∗). By induction hypothesis for n − 2, each has dimension ≥ 16 (n − 2)(n − 3)(n − 7), but 2

1 1 (n − 2)(n − 3)(n − 7) > n(n − 1)(n − 5), 6 6

for n ≥ 15, contradicting dim ([α] ↓ Sn−2 ) < 16 n(n − 1)(n − 5). This completes the proof of the lemma. 2 The following lemma is a special case of [14, Theorem 3.7]. Lemma 3.4. Let λ  n. If the Ferrers diagrams obtained from λ by removing 1 node from the right hand side from any row of the diagram so that the resulting diagram will still be a partition of (n − 1) are those of μ1 , . . . , μq , then ηλ (1) =

q n  μj f ημj (0). f λ j=1

For convenience, if λ = (n), we set dn = ηλ (0). By Theorem 1.1, dn = |Dn |. Lemma 3.5. For n ≥ 5, λ (n) (n − 1, 1) (1n ) (2, 1n−2 ) (n − 2, 2) (n − 2, 12 )

ηλ (1) ndn−1 0 (−1)n n(n − 2) 0 1 − (n−3) [dn−1 + (−1)n (n − 2)]  n − (n−1)(n−2) dn−1 + (−1)n−1 (n − 2)

(22 , 1n−4 ) (3, 1n−3 )

(−1)n−1 (n − 2)2 (−1)n n(n − 4)

Proof. These eigenvalues can be evaluated by using Theorem 2.2 (or Theorem 2.1) and Lemma 3.4. (a) η(n) (1) =

n f (n)

 (n−1) f η(n−1) (0) =

n 1

[1 · dn−1 ] = ndn−1 .

440

C.Y. Ku et al. / Linear Algebra and its Applications 493 (2016) 433–446

(b) By Theorem 2.2, η(n−2,1) (0) = −η(n−2) (0) − η(n−3) (0) = −dn−2 − dn−3 . Since dn−2 = (−1)n−2 + (n − 2)dn−3 , i.e., dn−3 = n−1 n−2 − (−1) +(n−1)d n−2

n−1 − dn−2

=

we have η(n−2,1) (0) =

(Theorem 2.2). Therefore,

n

η(n−1,1) (1) =

dn−2 +(−1)n−1 , n−2

 f (n−2,1) η(n−2,1) (0) + f (n−1) η(n−1) (0)

f (n−1,1)  

dn−1 n (n − 2) · − + 1 · dn−1 = n−1 n−2 = 0.

(c) By Theorem 2.1, η(1n−1 ) (0) = (−1)n−1 + (−1)n (n − 1) = (−1)n−2 (n − 2). Therefore, η(1n ) (1) =

n f

(1n )

n−1

f (1

)

 n 1 · (−1)n−2 (n − 2) = (−1)n n(n − 2). η(1n−1 ) (0) = 1

(d) By Theorem 2.1, η(2,1n−3 ) (0) = (−1)n−1 + (−1)n+1 (n − 1)η(1) (0) = (−1)n−1 . Therefore, η(2,1n−2 ) (1) =

n f

(2,1n−2 )

n−3

f (2,1

)

n−1

η(2,1n−3 ) (0) + f (1

)

 η(1n−1 ) (0)

n  (n − 2) · (−1)n−1 + 1 · (−1)n−2 (n − 2) n−1 = 0. =

  n−3 (e) By Theorem 2.2, η(n−3,2) (0) = η(n−3) (0) − 2η(n−4,1) (0) = dn−3 − 2 − dn−4 = n−2 n−4 dn−3 .

Therefore,

η(n−2,2) (1) = = = = = =

n f (n−2,2)

 f (n−3,2) η(n−3,2) (0) + f (n−2,1) η(n−2,1) (0)



 (n − 1)(n − 4) n − 2 dn−1 2n · dn−3 + (n − 2) · − n(n − 3) 2 n−4 n−2   (n − 1)(n − 2) 2 −dn−1 + dn−3 (n − 3) 2 

 2 (n − 1)(n − 2) dn−2 + (−1)n−1 −dn−1 + (n − 3) 2 n−2   (n − 1) (n − 1) 2 n −dn−1 + dn−2 − (−1) (n − 3) 2 2   1 (n − 1) 2 −dn−1 + (dn−1 + (−1)n ) − (−1)n (n − 3) 2 2

=−

1 [dn−1 + (−1)n (n − 2)] . (n − 3)

C.Y. Ku et al. / Linear Algebra and its Applications 493 (2016) 433–446

441

  n−2 (f) By Theorem 2.2, η(n−3,12 ) (0) = −η(n−3,1) (0) + η(n−4) (0) = − − dn−3 + dn−4 . Therefore, η(n−2,12 ) (1) =

f

 2 f (n−3,1 ) η(n−3,12 ) (0) + f (n−2,1) η(n−2,1) (0)

n (n−2,12 )



2n dn−2 (n − 2)(n − 3) = · dn−4 − − (n − 1)(n − 2) 2 n−3

 dn−1 + (n − 2) · − n−2   n−2 (n − 2)(n − 3) 2n −dn−1 + dn−2 + dn−4 = (n − 1)(n − 2) 2 2   n−2 n−2 2n n −dn−1 + dn−2 + (dn−3 + (−1) ) = (n − 1)(n − 2) 2 2   n−2 n−2 n−2 2n −dn−1 + dn−2 + dn−3 + (−1)n = (n − 1)(n − 2) 2 2 2   n−2 1 2n −dn−1 + dn−2 + dn−2 + (−1)n−1 = (n − 1)(n − 2) 2 2  n−2 (−1)n + 2   n−1 n−3 2n n −dn−1 + dn−2 + (−1) = (n − 1)(n − 2) 2 2   2n 1 n−3 −dn−1 + (dn−1 + (−1)n ) + (−1)n = (n − 1)(n − 2) 2 2   1 (−1)n 2n − dn−1 + (n − 2) = (n − 1)(n − 2) 2 2  n dn−1 + (−1)n−1 (n − 2) . =− (n − 1)(n − 2) (g) By Theorem 2.1, η(22 ,1n−5 ) (0) = (−1)n−2 η(1) (0) + (−1)n (n − 2)η(1,1) (0) = (−1)n (n − 2)η(1,1) (0) = (−1)n−1 (n − 2). Therefore, η(22 ,1n−4 ) (1) =

n

n−3

f (2,1

)

2

η(2,1n−3 ) (0) + f (2

,1n−5 )

 η(22 ,1n−5 ) (0)

f (22 ,1n−4 )   2n (n − 1)(n − 4) n−1 n−1 = (n − 2) · (−1) · (−1) + (n − 2) n(n − 3) 2

= (−1)n−1 (n − 2)2 . (h) By Theorem 2.1, η(3,1n−4 ) (0) = (−1)n−1 +(−1)n (n −1)η(2) (0) = (−1)n−1 +(−1)n (n − 1) = (−1)n (n − 2). Therefore

C.Y. Ku et al. / Linear Algebra and its Applications 493 (2016) 433–446

442

η(3,1n−3 ) (1) =

n f (3,1n−3 )

n−3

f (2,1

)

n−4

η(2,1n−3 ) (0) + f (3,1

)

 η(3,1n−4 ) (0)

  2n (n − 2)(n − 3) n−1 n = (n − 2) · (−1) · (−1) (n − 2) + (n − 1)(n − 2) 2 = (−1)n n(n − 4).

2

Lemma 3.6. (See [6, Lemma 2.5].) Let H be a graph on N vertices whose adjacency matrix A has eigenvalues η1 ≥ η2 ≥ . . . ≥ ηN , then N 

ηi2 = 2e(H),

i=1

where e(H) is the number of edges in H. Lemma 3.7. |S(n, 1)| = ndn−1 . Proof. Let Ai = {α ∈ Sn : α(i) = i and α(j) = j ∀j ∈ [n] \ {i}}. Note that the restriction of Ai on [n] \ {i} is the derangement of [n] \ {i}. Thus, |Ai | = n dn−1 . Since S(n, 1) = i=1 Ai and Ai ∩ Aj = ∅ for i = j, we have |S(n, 1)| = ndn−1 . 2 Lemma 3.8. Let n be positive integer such that n ≥ 6, and λ  n. If the dimension of the Specht module S λ , f λ ≥ 16 n(n − 1)(n − 5), then  |ηλ (1)| ≤ 6

dn−1 (n − 2)(n − 3)(n − 4)(n − 6)! . (n − 1)(n − 5)

Proof. By Theorem 1.1, Lemmas 3.6 and 3.7, 

2 f λ ηλ (1) = 2e (F(n, 1)) = n!|S(n, 1)| = n! (ndn−1 ) .

λn

This implies that 

n! (ndn−1 ) fλ  6 n! (ndn−1 ) ≤ n(n − 1)(n − 5)  dn−1 (n − 2)(n − 3)(n − 4)(n − 6)! . =6 (n − 1)(n − 5)

|ηλ (1)| ≤

2

C.Y. Ku et al. / Linear Algebra and its Applications 493 (2016) 433–446

443

4. Proof of Theorem 1.3 Lemma 4.1. For n ≥ 4, dn >

n! 3.

Proof. By Theorem 2.2, dn = (−1)n + ndn−1 . For n = 4, the lemma holds. Suppose n ≥ 5. Assume that the lemma holds for n − 1, i.e., dn−1 > (n−1)! . Since both sides are 3 (n−1)! integers, dn−1 ≥ 3 + 1. Therefore, dn ≥ −1 + ndn−1 ≥ −1 + n

n! n! (n − 1)! +1 ≥ + (n − 1) > . 3 3 3

Hence, the lemma follows. 2 Lemma 4.2. Let n ≥ 14 be a positive integer and λ  n. If the dimension f λ of the Specht module S λ is at least 16 n(n − 1)(n − 5), then   |ηλ (1)| < η(n−2,2) (1) . Proof. By Lemma 3.8,  |ηλ (1)| ≤ 6   By Lemma 3.5, η(n−2,2) (1) = it is sufficient to show that  6 ⇐⇒

dn−1 (n − 2)(n − 3)(n − 4)(n − 6)! . (n − 1)(n − 5)

1 (n−3)

[dn−1 + (−1)n (n − 2)] ≥

1 (n−3)

[dn−1 − (n − 2)]. So,

1 dn−1 (n − 2)(n − 3)(n − 4)(n − 6)! < [dn−1 − (n − 2)] (n − 1)(n − 5) (n − 3)

1 36dn−1 (n − 2)(n − 3)(n − 4)(n − 6)! 2 < [dn−1 − (n − 2)] (n − 1)(n − 5) (n − 3)2

⇐⇒

36dn−1 (n − 2)(n − 3)3 (n − 4)(n − 6)!  < (n − 1)(n − 5) d2n−1 − 2(n − 2)dn−1 + (n − 2)2

⇐⇒

36dn−1 (n − 2)(n − 3)3 (n − 4)(n − 6)! + 2dn−1 (n − 1)(n − 2)(n − 5)  < (n − 1)(n − 5) d2n−1 + (n − 2)2 .

Note that for n ≥ 7, (n − 2)(n − 3)3 (n − 4)(n − 6)! > (n − 2)(n − 5)(n − 3)3 (n − 6)! > 16(n − 2)(n − 5)(n − 3) > 2(n − 1)(n − 2)(n − 5).

(2)

444

C.Y. Ku et al. / Linear Algebra and its Applications 493 (2016) 433–446

Therefore, equation (2) follows provided that  37dn−1 (n − 2)(n − 3)3 (n − 4)(n − 6)! < (n − 1)(n − 5) d2n−1 + (n − 2)2 .

(3)

By Lemma 4.1,

   (n − 1)! + (n − 2)2 (n − 1)(n − 5) d2n−1 + (n − 2)2 > (n − 1)(n − 5) dn−1 3

(n − 1)! > (n − 1)(n − 5)dn−1 . 3 So, equation (3) follows provided that (n − 1)(n − 5)(n − 1)! > 111(n − 2)(n − 3)3 (n − 4)(n − 6)!, which is equivalent to (n − 1)2 (n − 5)2 > 111(n − 3)2 .

(4)

Finally, note that equation (4) holds for n = 14, 15 and for n ≥ 16, (n − 1)2 (n − 5)2 > (n − 3)2 (n − 5)2 ≥ 112 (n − 3)2 > 111(n − 3)2 . This completes the proof of the lemma. 2 Lemma 4.3. Let n ≥ 7 be a positive integer. Then   |ηλ (1)| < η(n−2,2) (1) ,   for λ ∈ (n − 2, 12 ), (1n ), (22 , 1n−4 ), (3, 1n−3 ) . Proof. By Lemma 3.5, it is sufficient to show the following two equations hold: 1 [dn−1 − (n − 2)] > n(n − 2); (n − 3) n 1 [dn−1 − (n − 2)] > [dn−1 + (n − 2)] . (n − 3) (n − 1)(n − 2)

(5) (6)

Note that equation (5) is equivalent to dn−1 > n(n − 2)(n − 3) + (n − 2),

(7)

and equation (7) holds provided that dn−1 > n(n − 1)(n − 2).

(8)

C.Y. Ku et al. / Linear Algebra and its Applications 493 (2016) 433–446

445

Next, equation (6) is equivalent to 2dn−1 > n(n − 2)(n − 3) + (n − 1)(n − 2)2 .

(9)

Note that equation (9) holds provided that equation (8) holds. Thus, it is sufficient to show that equation (8) holds. By Lemma 4.1, for n ≥ 7, dn−1 >

(n − 1)! ≥ 2(n − 1)(n − 2)(n − 3) 3 > n(n − 1)(n − 2).

This completes the proof of the lemma. 2 Proof of Theorem 1.3. It follows from Lemmas 3.2, 3.3, 3.5, 4.2 and 4.3 that for n ≥ 14, −

1 [dn−1 + (−1)n (n − 2)] (n − 3)

1 is the smallest eigenvalue of F(n, 1). Furthermore, ηλ (1) = − (n−3) [dn−1 + (−1)n (n − 2)] if and only if λ = (n − 2, 2). For 7 ≤ n ≤ 13, Theorem 1.3 can be verified by checking all the eigenvalues (see [14, Section 6]). This completes the proof of Theorem 1.3. 2

Let Γ be a graph. A set L of vertices of Γ is said to be an independent set in Γ if any two vertices in L are not adjacent to each other. Hoffman [11] observed the following useful bound on the size of an independent set in a regular graph (see also [7, Theorem 11]). Theorem 4.4 (Hoffman bound). Let Γ be a d-regular graph with n vertices and smallest eigenvalue τ . If I is an independent set in Γ, then |I| ≤

n 1−

d τ

.

Corollary 4.5. The size of a largest independent set in F(n, 1) is at most n![dn−1 + (−1)n (n − 2)] . (n2 − 3n + 1)dn−1 + (−1)n (n − 2) Proof. By Lemma 3.7, Theorems 1.3 and 4.4, if I is an independent set in F(n, 1), then |I| ≤

= =

n! 1−

ndn−1 1 − (n−3) [dn−1 +(−1)n (n−2)]

[dn−1 + (−1)n (n − 2)]n! dn−1 + (−1)n (n − 2) + n(n − 3)dn−1 (n2

n![dn−1 + (−1)n (n − 2)] . − 3n + 1)dn−1 + (−1)n (n − 2)

2

446

C.Y. Ku et al. / Linear Algebra and its Applications 493 (2016) 433–446

Acknowledgements We would like to thank the anonymous referee for the comments that helped us make several improvements to this paper. This project was supported by the Frontier Science Research Cluster, University of Malaya (RG367-15AFR). References [1] L. Babai, Spectra of Cayley graphs, J. Combin. Theory Ser. B 27 (1979) 180–189. [2] B.F. Chen, E. Ghorbani, K.B. Wong, Cyclic decomposition of k-permutations and eigenvalues of the arrangement graphs, Electron. J. Combin. 20 (4) (2013), #P22. [3] B.F. Chen, E. Ghorbani, K.B. Wong, On the eigenvalues of certain Cayley graphs and arrangement graphs, Linear Algebra Appl. 444 (2014) 246–253. [4] K. Day, A. Tripathi, Arrangement graphs: a class of generalized star graphs, Inform. Process. Lett. 42 (1992) 235–241. [5] P. Diaconis, M. Shahshahani, Generating a random permutation with random transpositions, Z. Wahrsch. Verw. Gebiete 57 (1981) 159–179. [6] D. Ellis, A proof of the Cameron–Ku conjecture, J. Lond. Math. Soc. 85 (2012) 165–190. [7] D. Ellis, E. Friedgut, H. Pilpel, Intersecting families of permutations, J. Amer. Math. Soc. 24 (2011) 649–682. [8] J.S. Frame, G. de B. Robinson, R.M. Thrall, The hook graphs of the symmetric group, Canad. J. Math. 6 (1954) 316–325. [9] W. Fulton, J. Harris, Representation Theory: A First Course, Springer-Verlag, 1991. [10] C.D. Godsil, B.D. McKay, Feasibility conditions for the existence of walk-regular graphs, Linear Algebra Appl. 30 (1980) 51–61. [11] A.J. Hoffman, On eigenvalues and colorings of graphs, in: Graph Theory and Its Applications, Academic Press, New York, 1970, pp. 79–91. [12] I.M. Isaacs, Character Theory of Finite Groups, Academic Press, New York, 1976. [13] R. Krakovski, B. Mohar, Spectrum of Cayley graphs on the symmetric group generated by transpositions, Linear Algebra Appl. 437 (2012) 1033–1039. [14] C.Y. Ku, T. Lau, K.B. Wong, Cayley graph on symmetric group generated by elements fixing k points, Linear Algebra Appl. 471 (2015) 405–426. [15] C.Y. Ku, T. Lau, K.B. Wong, Largest independent sets of certain regular subgraphs of the derangement graph, J. Algebraic Combin. (2015), http://dx.doi.org/10.1007/s10801-015-0656-4, in press. [16] C.Y. Ku, D.B. Wales, Eigenvalues of the derangement graph, J. Combin. Theory Ser. A 117 (2010) 289–312. [17] C.Y. Ku, K.B. Wong, Solving the Ku–Wales conjecture on the eigenvalues of the derangement graph, European J. Combin. 34 (2013) 941–956. [18] A. Lubotzky, Discrete Groups, Expanding Graphs, and Invariant Measures, Birkhäuser Verlag, Basel, 1994. [19] M. Ram Murty, Ramanujan graphs, J. Ramanujan Math. Soc. 18 (2003) 1–20. [20] P. Renteln, On the spectrum of the derangement graph, Electron. J. Combin. 14 (2007), #R82. [21] B.E. Sagan, The Symmetric Group, Grad. Texts in Math., vol. 203, Springer-Verlag, New York, 2001. [22] J.P. Serre, Linear Representations of Finite Groups, Springer-Verlag, New York, 1977. [23] R.P. Stanley, Enumerative Combinatorics, vol. 2, Cambridge University Press, Cambridge, 1999.