On the second largest eigenvalue of some Cayley graphs of the Symmetric Group

Let $S_n$ and $A_{n}$ denote the symmetric and alternating group on the set $\{1,...,\, n\},$ respectively. In this paper we are interested in the second largest eigenvalue $\lambda_{2}(\Gamma)$ of the Cayley graph $\Gamma={\rm Cay}(G,H)$ over $G=S_{n}$ or $A_{n}$ for certain connecting sets $H.$ Let $1<k\leq n$ and denote the set of all $k$-cycles in $S_{n}$ by $C(n,k).$ For $H=C(n,n)$ we prove that $\lambda_{2}(\Gamma)=(n-2)!$ (when $n$ is even) and $\lambda_{2}(\Gamma)=2(n-3)!$ (when $n$ is odd). Further, for $H=C(n,n-1)$ we have $\lambda_{2}(\Gamma)=3(n-3)(n-5)!$ (when $n$ is even) and $\lambda_{2}(\Gamma)=2(n-2)(n-5) !$ (when $n$ is odd). The case $H=C(n,3)$ has been considered in X. Huang and Q. Huang, 'The second largest eigenvalue of some Cayley graphs on alternating groups', J. Algebraic Combinatorics, 50(2019), $99-111$. Let $1\leq r<k<n$ and let $C(n,k;r) \subseteq C(n,k)$ be set of all $k$-cycles in $S_{n}$ which move all the points in the set $\{1,2,...,r\}.$ That is to say, $g=(i_{1},i_{2},...,\, i_{k})(i_{k+1})\dots(i_{n})\in C(n,k;r)$ if and only if $\{1,2,...,\,r\}\subset \{i_{1},i_{2},...,\, i_{k}\}.$ Our main result concerns $\lambda_{2}(\Gamma)$, where $\Gamma={\rm Cay}(G,H)$ with $H=C(n,r+1,r)$ for $2\leq r\leq n-2.$ Suppose that $G=S_{n}$ if $r$ is odd and $G=A_{n}$ if $r$ is even. We prove that $\lambda_{2}(\Gamma)=r!(n-r-1)$. The cases with $H=C(n,3;1)$ and $H=C(n,3;2)$ have been considered in the same paper of X. Huang and Q. Huang.


Introduction
Let Γ be a finite undirected graph and let A denote its adjacency matrix. The eigenvalues of this matrix, together with their multiplicities, are an important invariant of the graph. Since A is symmetric all eigenvalues are real. For a regular graph of degree d it is well-known that the largest eigenvalue of A is λ 1 (Γ) = d. The second largest eigenvalue, denoted by λ 2 (Γ), plays an important role in many theoretical and practical applications of graph theory, from geometry to computer science. The literature on the second largest eigenvalue is extensive, overviews and further references can be found in [1,3,5,11,12,13]. For regular graphs the spectral gap λ 1 (Γ) − λ 2 (Γ) is known as the algebraic connectivity of Γ.
In this paper we are concerned with the second largest eigenvalue of certain Cayley graphs Cay(G, H) where G is a symmetric or alternating group and where H a connecting set in G.
(In particular, we have H = H −1 = {h −1 | h ∈ H}, the identity element of G does not belong to H, and H is not contained in any proper subgroup of G.) For the definition of Cayley graphs see Section 2.4. References to recent work on the second largest eigenvalue of Cayley graphs over symmetric groups can be found in [6]. Even for symmetric groups an explicit computation of it appears to be one of the most challenging problem in the area. 1 To state our results let S n = Sym(1, . . . , n) and A n be the symmetric and alternating group on the set {1, . . . , n}, respectively. For 1 < k ≤ n a k -cycle in S n is a permutation of the shape g = (i 1 , i 2 , . . . , i k )(i k+1 ) . . . (i n ) with |{i 1 , i 2 , . . . , i k }| = k. The set of all k -cycles in S n is denoted by C(n, k). Theorem 1.1. Let Γ = Cay(G, H) with H = C(n, n) for n > 4 and G = S n when n is even and G = A n when n is odd. Then λ 1 (Γ) = (n − 1)!. Furthermore, λ 2 (Γ) = (n − 2)! when n is even and λ 2 (Γ) = 2(n − 3)! when n is odd. Theorem 1.2. Let Γ = Cay(G, H) with H = C(n, n−1) for n > 4 and G = S n when n is odd and G = A n when n is even. Then λ 1 (Γ) = n(n − 2)!. Furthermore, λ 2 (Γ) = 3(n − 3)(n − 5)! when n is even and λ 2 (Γ) = 2(n − 2)(n − 4)! when n is odd.
An essential feature of the Cayley graphs in these two theorems is that the connecting set H is invariant under conjugation by G. In fact, C(n, k) is a conjugacy class of S n . If k = n or n − 1, the above results are deduced from some analysis of the group characters in a relatively straightforward way. For arbitrary k computing the second largest eigenvalue for Cay(G, C(n, k)) is an open problem. For k = 3 this is solved in [6,Theorem 3.4], for k = 2 see [9, Lemma 3].
In this theorem, by contrast, the connecting set H is not invariant under S n . However, we make essential use of the fact that H is invariant under a subgroup isomorphic to F = S r × S n−r . For Cayley graph Γ = Cay(G, H) with H = C(n, 3; 1) and H = C(n, 3; 2) the value of λ 2 (Γ) has been determined in [6,Theorems 3.2 and 2.3]. This values is also known for H = C(n, 2; 1) see [11,Section 8.1].
Notation: All groups considered here are finite. Further, all modules and representations are over the field C of complex numbers. By CG we denote the group algebra of the group G over C . If H ⊂ G then |H| is the number of elements of H and H + is the sum of all h ∈ H, as an element of CG. A sentence such as 'L is a G-module' means that L is a CG-module. If L is a CG-module and X is a subgroup of G, we write L| X for the restriction of L to X . The trivial G-module and its character is denoted by 1 G . We assume that the reader is familiar with general notions and elementary facts of finite group theory, including the representation theory of S n and A n .

Preliminaries
We collect several well-known facts and prerequisites for the paper. All graphs are finite, undirected and without multiple edges.

Graphs and Eigenvalues. Let Γ be a graph on the vertex set
The group of all automorphisms of Γ is denoted by Aut(Γ).
The adjacency matrix A = (a u,v ) of Γ is given by a u,v = 1 if u ∼ v and a u,v = 0 otherwise. Then A determines Γ. The eigenvalues of Γ are, by definition, the eigenvalues of A. These are real since A is symmetric. (2) Γ is bipartite if and only if −d is an eigenvalue of A; (3) d ≥ |λ| for all eigenvalues λ of A. If Γ is bipartite and if λ is an eigenvalue of A then −λ is an eigenvalue of A with the same multiplicity as λ.

Equitable partitions.
Let Γ be a graph with vertex set V. Let P be a partition As usual, a second partition P ′ is a refinement of P if every class of P is some union of classes of P ′ .
For any graph Γ examples of equitable partitions arise from the orbits of a group of automorphisms of Γ, see Lemma 2.3 below. For this reason equitable partitions are also called generalized orbits. For incidence graphs of geometries also the term tactical decomposition is used, see Dembowski's book [4].
Let CV be the vector space with basis V over C. Then A can be viewed as a linear transformation of CV , and then Av = x∈N (v) x for every v ∈ V . Let (·, ·) be the standard bilinear form on CV defined by (u, v) = δ u,v (the Kronecker delta) for u, v ∈ V . Let V = V 1 ∪ · · · ∪ V k be a partition P of V . Set w i = u∈V i u for i = 1, . . . , k . Then (w i , Av) = |N (v) ∩ V i | (as (w i , u) = 1 for u ∈ V if and only if u ∈ V i ). Let CP be the vector space with basis w 1 , . . . , w k . In this notation we have: Lemma 2.2. Let Γ be a graph with vertex set V and adjacency matrix A, and let P be a . Furthermore, if P is equitable then B(P ) is the matrix of the restriction of A to CP for the basis w 1 , . . . , w k . In particular, all eigenvalues of B are eigenvalues of Γ. If Γ is regular of degree d then d is an eigenvalue of B.
In addition, if P ′ is an equitable refinement of P then the eigenvalues of B(P ) are eigenvalues of B(P ′ ).
Proof: For i = 1, . . . , k we have To be in CP the right hand side must be constant on every V j , that is, |N (v)∩V i | = |N (v ′ )∩V i | for every j ∈ {1, . . . , k} and every v, v ′ ∈ V j . The converse and the second assertion of the lemma follows from (1) as well. Finally, if Γ is regular of degree d then A(w 1 + · · · + w k ) = A v∈V v = d v∈V v = d(w 1 + · · · + w k ), so d is an eigenvalue of A on CP .
Let V = V ′ 1 ∪ · · · ∪ V ′ ℓ be a refinement P ′ of P and let as above w ′ 1 , . . . , w ′ l be a basis of CP ′ . Then every w j (j = 1, . . . , k ) is a linear combination of w ′ 1 , . . . , w ′ ℓ , so CP ⊂ CP ′ , whence the additional statement. ✷ Lemma 2.3. Let Γ be a graph and G a group of automorphisms of Γ. Let V 1 , ..., V k be the orbits of G on the vertex set V of Γ.
2.3. Eigenvalue Inequalities. We denote the distinct eigenvalues of a graph Γ by Upper and lower bounds on eigenvalues of a symmetric matrix over the reals can be obtained from a theorem of Hermann Weyl [15], see also [3, Theorem 2.8.1].
Let v be a vertex of Γ and let Γ ′ be obtained from Γ by deleting all edges incident with v.
Denote the adjacency matrix of Γ ′ by A ′ and let γ ′ 1 ≥ γ ′ 2 ≥ · · · ≥ γ ′ m be all eigenvalues of Γ ′ , with repetitions. Next let Σ be the 'star' at v with d rays, where d is the degree of v, and with m − d − 1 isolated vertices. Denote its adjacency matrix by S. The eigenvalues of the star are σ 1 = √ d > σ 2 = 0 = · · · = σ m−1 > σ m = − √ d, by a simple computation. Then A = A ′ + S and from Theorem 2.5 we obtain the following Corollary 2.6. Let v be a vertex of the connected graph Γ and let Γ ′ be obtained by deleting all edges incident with v. Let λ 1 > λ 2 and λ ′ 1 > λ ′ 2 be the largest and second largest eigenvalues of Γ and Γ ′ , respectively. Then Accounting for the isolated vertex v, the spectrum of the graph Γ[V \ {v}] induced on V \ {v} is obtained from the spectrum of Γ ′ by removing one eigenvalue λ ′ = 0. The first part of the corollary is therefore an instance of the well-known interlacing theorem, see [3, Theorem 2.5.1]. However, the second part of the corollary appears to be new.

Cayley Graphs.
Let G be a group with identity element 1 or 1 G and let H be a subset Thus, such a generalization is not essential. However, for the study of eigenvalues of Cay(G 1 , H) by means of representation theory one sometimes prefers to deal with the adjacency matrix of Γ(G, H) if the representation theory of G is simpler than that of G 1 . In Section 6 we observe this for G = S n and G 1 = A n .
There is no harm in identifying g ∈ G with its matrix in the regular representation of G, of size |G| × |G|. A key fact for adjacency matrices of Cayley graphs then is that h∈H h, the sum over h ∈ H of the matrices just defined, is the adjacency matrix of Cay(G, H). Thus A(G, H) = H + under this identification. (Recall that we define H + as an element of the group algebra of G over C .) This is well known and explained, for instance, in [14, p. 384]. In other words, A is the image of H + in the regular representation of the group algebra. Note that A is a linear transformation of the vector space CV = CG, the C -span of G.  g) :

Some general results
In this section we discuss representation theoretical aspects of eigenvalue problems for Cayley graphs. Let G be a finite group, H ⊂ G a connecting subset, and put H + = h∈H as an element in the group algebra CG of G over C. If ρ is the regular representation of G then A(G, H) = ρ(H + ). If we identify g with ρ(g) then A(G, H) = H + as discussed in Section 2.4. Thus, if φ is any representation of G then φ(H + ) is meaningful, and the eigenvalues of φ(H + ) are eigenvalues of Γ. We think that this convention makes the exposition more transparent.
By standard results, ρ is a direct sum of all irreducible representations φ of G, each occurring with multiplicity dim φ. It follows that the set of all eigenvalues of Cay(G, H) (disregarding the multiplicities) is the union, over the irreducible representations φ of G, of the eigenvalues of φ(H + ). See [10, Proposition 7.1] or elsewhere.
If φ 0 is the trivial representation of G then, obviously, φ 0 (H + ) = |H|. Conversely, if |H| is an eigenvalue of φ(H + ), where φ is an irreducible representation of G, then φ = φ 0 by Theorem 2.1 (as φ 0 occurs in ρ with multiplicity 1; here we use that H is a connecting set). (1) If H is G-stable (that is, gHg −1 = H for every g ∈ G) then the restriction H + to M acts as a scalar matrix µ·Id.
Then H + acts scalarly on L.
(ii) By (2), H + i stabilizes L and acts on it as ν i · Id for i = 1, 2. As H + = H + 1 + H + 2 , the claim follows. (2) If G = S n , G 1 = S n−1 then every irreducible G-module M is multiplicity free as G 1 -module (that is, every irreducible constituent of M | S n−1 occurs exactly once). This follows from the branching rule expressed in terms of Young diagrams (we cannot get the same diagram by removing distinct single boxes from a given Young diagram. This remains true for alternating groups.) In addition, if C is a conjugacy class of S n then C ∩ S n−1 is either empty or a single conjugacy class of G 1 , as two permutations x, y ∈ G 1 are conjugate in S n if and only if they are conjugate in G 1 . In the following we view Sym(1, . . . , k) as the largest subgroup of S n which fixes all i ∈ {1, . . . , k}, etc. Proof. Set F 1 = Sym(1, . . . , k)×Id and F 2 = Id ×Sym(k +1, . . . , n). As H is F -invariant, H + commutes with F in the group algebra CS n and hence in End M . We have M | where T is a trivial F -module of dimension 2. By Schur's lemma, H + acts scalarly on L 1 , L 2 , and hence H + has at most four distinct eigenvalues on M .
One observes that if k = 1 then L 0 = 0 and H + has at most three distinct eigenvalues on M .

Symmetric group: the largest conjugacy classes
For basic definitions concerning the characters of symmetric and alternating groups see [7,8].
Let G = S n and let 1 ≤ k ≤ n. We denote the set of all k -cycles in G by C(n, k). Since two permutations are conjugate to each other in G if and only if they have the same cycle type we have |C(n, k)| = |G : C G (g)| when g is any k -cycle and C G (g) denotes the centralizer of g in G. In particular, |C(n, n)| = (n − 1)! and |C(n, n − 1)| = n(n − 2)!. By the same argument, the enumeration of permutations by cycle type shows that all other conjugacy class sizes in G are smaller if n > 4. It is well-known that the smallest subgroup of S n containing C(n, k) is S n if k is even and A n if k is odd.
In this section we compute the second largest eigenvalue of H + when H = C(n, n) or H = C(n, n − 1). For this we first quote a well-known result for the cycles of length n in representations of S n [7, Lemma 21.4].
By Lemma 2.7 we have:

The natural S n -module and equitable partitions
Let G = S n and let K be the stabilizer of some point in {1, . . . , n}. Then the set G/K of cosets of K in G and {1, . . . , n} are isomorphic as G-sets. The associated permutation module over C is the natural module for G, denoted M = 1 G K . This is probably the most important S n -module. Moreover, we expect that in a majority of cases the second largest eigenvalue of Cayley graphs over S n occurs on M (although we have no means to justify this).
For 1 ≤ r < k ≤ n let C(n, k; r) ⊆ C(n, k) be the set of all k -cycles in S n which move all the points in the set {1, 2, . . . , r}. That is to say, g = (i 1 , i 2 , . . . , i k )(i k+1 ) . . . (i n ) ∈ C(n, k; r) if and only if {1, 2, . . . , r} ⊂ {i 1 , i 2 , . . . , i k }. Clearly, |C(n, k; r)| = (k − 1)! n−r k−r . Lemma 5.1. Let 1 ≤ r < k ≤ n and let X be the smallest subgroup of S n containing C(n, k; r). Then X = S n if k is even, and X = A n if k is odd.
Proof. This holds when k = n since C(n, n; r) = C(n, n) for all r and so it suffices to show that X ⊇ Alt n for 1 ≤ r < k < n. Since for every i ∈ {2, . . . , n} there is some h ∈ C(n, k; r) with h(i) = 1 it follows that X is transitive on {1, 2, . . . , n}. If i < n we may assume additionally that h(n) = n since k < n. Hence the stabilizer of n in X is transitive on {1, 2, . . . , n − 1} and so X is doubly transitive. The result follows if k = 2 or = 3, since a double transitive group containing a 2-or 3-cycle contains Alt n . For k ≥ 4 consider the elements h = (1, 2, . . . , k − 3, k − 2, k − 1, k) and h ′ = (k − 1, k, k − 2, k − 3, . . . , 2, 1). Then In the remainder of this section let r, k, G and H be as in the theorem and put Γ = Cay(G, H). We exhibit two equitable partitions of Γ so that the associated eigenvalues belong to {µ 1 , µ 2 , µ 3 , µ 4 }. The proof is completed at the end of the section.
This gives the matrix above after some further transformations. One easily checks that the eigenvalues of B 1 are as stated.
If we take x = n − r − 1, y = 1, z = r − 1 then the equality holds. In addition, the equality fails for any other choice of {x, y, z} ∈ {1, n − r − 1, r − 1}. Whence the result.
Corollary 6.2. Let M ′ be an S n -module whose composition factor dimensions are at most n − 1. Then the second largest eigenvalue of H + on M ′ does not exceed r!(n − r − 1).
Proof. Let ρ be an irreducible characters of S n of degree at most n − 1. Then ρ is well known to be a constituent either of M or M ⊗ σ , where σ denotes the sign representation of S n . It follows that the eigenvalues of H + on M ′ are in the list of Lemma 3.2 if r is even, and also their negatives, if r is odd. So the result follows as r!(n − r − 1) > (r − 1)!(n − r). Lemma 6.3. Let G = S n and let L be an irreducible G-module. Then there is a basis of L such that the matrices g + g −1 (g ∈ G) are symmetric. Consequently, if H ⊂ G is a subset such that H = H −1 then the matrix of H + on L is symmetric.
Proof. It is well known that there is a basis of L such that all matrices g ∈ G are orthogonal, that is, if g L is the matrix of g on L then g T L = (g L ) −1 = (g −1 ) L , where (g L ) T is the transpose of g L . Then g L + g −1 L = g L + (g L ) T , whence the claim. Lemma 6.4. Let L be an irreducible S n -module and λ an eigenvalue of H + on L. Then λ ≤ |H| and λ = |H| implies dim L = 1.
Proof. If H is connecting then the lemma follows from Theorem 2.1. Indeed, λ = |H| if L is the trivial module, and the trivial module has multiplicity 1 in CS n . If H is not a connecting set then H is a connecting set of A n (Lemma 5.1), and the same argument is valid if L is an A n -module. So |H| is an eigenvalue of H + on L if and only if the restriction of L to A n contains the trivial A n -module. This is well known to imply that dim L = 1.