The second eigenvalue of some normal Cayley graphs of high transitive groups

Let $\Gamma$ be a finite group acting transitively on $[n]=\{1,2,\ldots,n\}$, and let $G=\mathrm{Cay}(\Gamma,T)$ be a Cayley graph of $\Gamma$. 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 $\max_{\tau\in T}|\mathrm{supp}(\tau)|\leq 5$, where $\mathrm{supp}(\tau)$ is the set of points in $[n]$ non-fixed by $\tau$.

Let Γ be a finite group, and let T be a subset of Γ such that e ∈ T (e is the identity element of Γ) and T = T −1 . The Cayley graph Cay(Γ, T ) of Γ with respect to T (called connection set) is defined as the undirected graph with vertex set Γ and edge set {{γ, τ γ} | γ ∈ Γ, τ ∈ T }. Clearly, Cay(Γ, T ) is a regular graph which is connected if and only if T is a generating subset of Γ. A Cayley graph Cay(Γ, T ) is called normal if T is closed under conjugation.
Let S n be the symmetric group on [n] = {1, 2, . . . , n} with n ≥ 3, and T a subset of S n consisting of transpositions. The transposition graph Tra(T ) of T is defined as the graph with vertex set {1, 2, . . . , n} and with an edge connecting two vertices i and j if and only if (i, j) ∈ T . It is known that T can generate S n if and only if Tra(T ) is connected [21]. In 1992, Aldous [1] (see also [9,19]) conjectured that the spectral gap of Cay(S n , T ) is equal to the algebraic connectivity (second least Laplacian eigenvalue) of Tra(T ). Earlier efforts of several researchers solved various special cases of Aldous' conjecture. For instance, Diaconis and Shahshahani [15], and Flatto, Odlyzko and Wales [18] confirmed the conjecture for Tra(T ) being a complete graph and a star, respectively; Handjani and Jungreis [22] confirmed the conjecture for Tra(T ) being a tree; Friedman [19] proved that if Tra(T ) is a bipartite graph then the spectral gap of Cay(S n , T ) is at most the algebraic connectivity of Tra(T ); Cesi [9] confirmed the conjecture for Tra(T ) being a complete multipartite graph. At last, Caputo, Liggett and Richthammer [7] completely confirmed the conjecture in 2010, their proof is an ingenious combination of two ingredients: a nonlinear mapping in the group algebra CS n which permits a proof by induction on n, and a quite complicated estimate named the octopus inequality (see also [10] for a self-contained algebraic proof). Very recently, Cesi [11] proved an analogous result of Aldous' conjecture (now theorem) for the Weyl group W (B n ). Most of the above results rely heavily on the representation theory of the symmetric group S n .
The second eigenvalues of Cayley graphs of the symmetric group S n or the alternating groups A n have been determined also for some special generators that are not transpositions. For 1 ≤ i < j ≤ n, let r i,j ∈ S n be defined as r i,j = 1 · · · i − 1 i i + 1 · · · j − 1 j j + 1 · · · n 1 · · · i − 1 j j − 1 · · · i + 1 i j + 1 · · · n .
In [8], Cesi proved that the second eigenvalue of the pancake graph P n = Cay(S n , {r 1,j | 2 ≤ j ≤ n}) is equal to n − 2. 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]). Suppose that Γ is a finite group acting transitively on [n] and let G = Cay(Γ, T ). In the present paper, we first show that, for each i ∈ [n], the left coset decomposition of Γ with respect to the stabilizer subgroup Γ i is an equitable partition of G, and all these equitable partitions share the same quotient matrix B Π . Based on this fact, we also prove that those eigenvalues of G not belonging to B Π can be bounded above by the sum of second eigenvalues of some subgraphs of G. Now suppose further that G is connected and normal, and that the action of Γ on [n] is of high transitivity. Using the previous result, we reduce the problem of proving λ 2 (G) = λ 2 (B Π ) to that of verifying the result for some smaller graphs. This leads to a recursive procedure for determining the second eigenvalue of G. As applications, we determine the second eigenvalues of a majority of connected normal Cayley graphs of S n with max τ ∈T |supp(τ )| ≤ 5 (see Theorem 4.1 and Table 2), where supp(τ ) is the set of points in [n] non-fixed by τ . There are 56 families of such graphs, and we determine the second eigenvalues for 41 families of them. In the process, we also determine the second eigenvalues of some subgraphs (over one hundred families) of these 41 families of normal Cayley graphs. From these results we can determine the spectral gap of Cay(S n , {(p, q) | 1 ≤ p, q ≤ n}) (previously done by Diaconis and Shahshahani [15]) and Cay(S n , {(1, q) | 2 ≤ q ≤ n}) (previously obtained by Flatto, Odlyzko and Wales [18,Theorem 3.7]). We show that a recent conjecture of Dai [14] is true as a consequence of Aldous' theorem and we discuss some related questions and open problems.

Main tools
Let G be a graph on n vertices. The vertex partition Π : is the quotient matrix of G with respect to Π, and the n×q matrix χ Π whose columns are the characteristic vectors of V 1 , . . . , V q is the characteristic matrix of Π. Lemma 2.1 (Brouwer and Haemers [5], p. 30; Godsil and Royle [21], pp. 196-198). Let G be a graph with adjacency matrix A(G), and let Π : V (G) = V 1 ∪ V 2 ∪ · · · ∪ V q be an equitable partition of G with quotient matrix B Π . Then the eigenvalues of B Π are also eigenvalues of A(G). Furthermore, A(G) has the following two kinds of eigenvectors: (i) the eigenvectors in the column space of χ Π , and the corresponding eigenvalues coincide with the eigenvalues of B Π ; (ii) the eigenvectors orthogonal to the columns of χ Π , i.e., those eigenvectors that sum to zero on each block For regular graphs, we have the following useful result.
Theorem 2.2. Let G be a r-regular graph, and let λ (λ = r) be an eigenvalue of G. If G has an eigenvector f with respect to λ and a vertex partition Π : where G 1 is the (r − r 1 )-regular graph obtained from G by removing all edges in Proof. By assumption, the induced subgraphs G[V i ] share the same degree r 1 , so G 1 is (r − r 1 )-regular because G is r-regular. Also, the eigenvector f of λ sums to zero on V i for each i.
. By the Rayleigh quotient, we obtain For the first term, we have where f | V i is the restriction of f on V i , 1 V i is the all ones vector on V i , and the second inequality follows from x∈V i f (x) = 0 (1 ≤ i ≤ q). For the second term, since G 1 is regular and f is orthogonal to the all ones vector 1, we have Combining (1), (2) and (3), we conclude that and the result follows.
If the partition Π : V (G) = V 1 ∪ V 2 ∪ · · · ∪ V q is exactly an equitable partition of G with quotient matrix B Π , then the eigenvectors of G with respect to those eigenvalues other than that of B Π must sum to zero on each V i by Lemma 2.1. From Theorem 2.2 one can immediately deduce the following result.
is an equitable partition of G whose quotient matrix B Π has constant diagonal entries. Then, for any eigenvalue λ of G that is not that of B Π , we have Here we give an example to show how to use the result of Corollary 2.3.

Example 1.
Let H 1 , H 2 be two connected k-regular graphs on n vertices. Let G be the graph (not unique) obtained from H 1 ∪ H 2 by adding some new edges between H 1 and H 2 such that these edges form a r-regular bipartite graph G 1 (G 1 is easy to construct, cf. [26], Lemma 3.2). Clearly, G is a connected (k + r)-regular graph. Let V 1 and V 2 be the vertex subsets of G corresponding to H 1 and H 2 , respectively. Then V (G) = V 1 ∪ V 2 is clearly an equitable partition of G with quotient matrix Since λ 2 (G 1 ) ≤ r, each eigenvalue of G not belonging to B Π is bounded above by max{λ 2 (H 1 ), λ 2 (H 2 )}+r according to Corollary 2.3. As λ 2 (B Π ) = k−r, we conclude that Note that the above bounds could be tight. Take H 1 = H 2 = Q n , the n-dimensional hypercube, and let G be the graph (not unique) obtained from H 1 ∪ H 2 by adding a perfect matching between H 1 and H 2 (such graphs contain the (n + 1)-dimensional locally twisted cubes, cf. [33]). Since λ 2 (Q n ) = n − 2 (cf. [5], p. 19), we have and thus λ 2 (G) = n−1, which attains the lower bound. Also, the Cartesian product C n K 2 , which can be regarded as the graph obtained by adding a perfect matching between two copies of C n , has second eigenvalue 2 cos 2π n + 1 = λ 2 (C n ) + 1, and so attains the upper bound. By using Theorem 2.2, in what follows, we focus on providing upper bounds for some special eigenvalues of Cayley graphs. Before doing this, we need to do some preparatory work. First of all, we give the following useful result, which suggests that each Cayley graph has an equitable partition derived from left coset decomposition. Proof. Suppose that Π : Γ = γ 1 Θ ∪ γ 2 Θ ∪ · · · ∪ γ k Θ is the left coset decomposition of Γ with respect to Θ, where k = |Γ|/|Θ| and γ 1 , . . . , γ k are the representation elements. Clearly, Π is a vertex partition of Cay(Γ, T ). For any γ ∈ γ i Θ, we have γ = γ i θ for some θ ∈ Θ, and therefore which is independent on the choice of γ ∈ γ i Θ. Thus Π is exactly an equitable partition of Cay(Γ, T ), and the result follows.
Let Ω be a nonempty set, and let Γ be a group acting on Ω. We say that the action of Γ on Ω (|Ω| ≥ s) is s-transitive if for all pairwise distinct x 1 , . . . , x s ∈ Ω and pairwise distinct y 1 , . . . , y s ∈ Ω there exists some γ ∈ Γ such that x γ i = y i for 1 ≤ i ≤ s. Clearly, a s-transitive action is always t-transitive for any t < s. In particular, we say that the action is transitive if it is 1-transitive. As usual, we denote by Γ x = {γ ∈ Γ | x γ = x} the stabilizer subgroup of Γ with respect to x ∈ Ω. Now suppose that Γ is a finite group acting transitively on [n] = {1, 2, . . . , n}. For each fixed i ∈ [n], we have |Γ|/|Γ i | = n by the orbit-stabilizer theorem, and furthermore, we see that Γ has left coset decomposition where γ j,i is an arbitrary element in Γ that maps j to i and for all j ∈ [n]. Clearly, |Γ j,i | = |Γ i | = |Γ|/n. Let G = Cay(Γ, T ) be a Cayley graph of Γ. According to Lemma 2.4, for each i ∈ [n], the left coset decomposition Π i given in (4) is an equitable partition of G with quotient matrix is exactly the number of elements in T mapping t to s. Since b st = |T ∩ Γ t,s | is independent on the choice of i, all the equitable partitions Π i share the same quotient matrix. For this reason, we use B Π instead of B Π i . Also, by counting the edges between Γ s,i and Γ t,i in two ways, we obtain b st ·|Γ s,i | = b ts ·|Γ t,i |, which implies that b st = b ts because |Γ s,i | = |Γ t,i | = |Γ|/n. Therefore, B Π = (b st ) n×n is symmetric. For any fixed k ∈ [n], we also can partition the vertex set of G as another form which is exactly the right coset decomposition of Γ with respect to Γ k . In general, Π ′ k is not an equitable partition of G. As in Theorem 2.2, we can decompose the Proof. For (i), the corresponding isomorphism can be defined as Clearly, φ is one-to-one and onto. Furthermore, we have and so (i) follows. Now we consider (ii). Clearly, G 1 [Γ k,i ] is an empty graph for all i ∈ [n]. For any γ k,i γ ∈ Γ k,i = γ k,i Γ i and γ k,j γ ′ ∈ Γ k,j = γ k,j Γ j (i = j), we have {γ k,i γ, γ k,j γ ′ } ∈ E(G 1 ) if and only if γ k,j γ ′ (γ k,i γ) −1 ∈ T , which is the case if and This proves (ii). Now we are in a position to give the main result of this section, which provides upper bounds for some special eigenvalues of Cayley graphs.
where Γ k is the stabilizer subgroup of Γ with respect to k.
Proof. From the above arguments, it suffices to prove the second part of the theorem. Let f be an arbitrary eigenvector of G with respect to λ. Since Π i is an equitable partition of G for each i, we see that f must sum to zero on Γ j,i for all i, j ∈ [n] by Lemma 2.1. For any fixed k ∈ [n], let Π ′ k be the vertex partition of G given in (6).
In particular, we have that f sums to zero on Γ k,i for all i ∈ [n]. By Lemma 2.5, all these induced subgraphs G[Γ k,i ] (i ∈ [n]) are isomorphic to Cay(Γ k , T ∩ Γ k ), and so share the same degree |T ∩ Γ k |. Let G 1 be the graph obtained from G by removing all edges in ∪ n i=1 E(G[Γ k,i ]). Note that G 1 ∼ = Cay(Γ, T \ (T ∩ Γ k )) again by Lemma 2.5. Then, by applying Theorem 2.2 to the vertex partition Π ′ k , we obtain By the arbitrariness of k ∈ [n], our result follows.
It is worth mentioning that Theorem 2.6 provides for us a recursive method to determine the second eigenvalue of the connected Cayley graph G = Cay(Γ, T ). Indeed, by Lemma 2.1, all eigenvalues of B Π are also that of G, so we have then we may conclude that λ 2 (G) = λ 2 (B Π ) by Theorem 2.6. Thus the problem is reduced to determining the exact value of λ 2 (Cay(Γ k , T ∩ Γ k )) and λ 2 (Cay(Γ, T \ (T ∩ Γ k ))), which reminds us that the way of induction could be applied.
In the next section, we shall see that if Γ and T satisfy some additional conditions then the problem of proving λ 2 (G) = λ 2 (B Π ) can be reduced to that of verifying the result for some small graphs.

Normal Cayley graphs
For a finite group Γ, the conjugacy class of γ ∈ Γ is defined as the set C γ = {σ −1 γσ | σ ∈ Γ}. Recall that a Cayley graph Cay(Γ, T ) is said to be normal if T is closed under conjugation, that is, T is the disjoint union of some conjugacy classes of Γ. It is well known that the eigenvalues of a normal Cayley graph can be expressed in terms of the irreducible characters of Γ.
However, it is often difficult to identify the second eigenvalues of normal Cayley graphs from Theorem 3.1. In this section, by using Theorem 2.6, we reduce the problem of determining the second eigenvalues of normal Cayley graphs of high transitive groups to that of verifying the result for some smaller graphs.
From now on, we always assume that Γ acts transitively on [n], and that G = Cay(Γ, T ) is a connected normal Cayley graph of Γ, i.e., T is a generating subset of Γ which is also closed under conjugation. In order to use Theorem 2.6 recursively, we set T 0 = T , G 0 = Cay(Γ, T 0 ) = G, and for k = 1, 2, . . . , n, we define We see that both G k and H k are subgraphs of G k−1 , and furthermore, by regarding T k−1 as T in Lemma 2.5, we have can be decomposed into that of G k and n copies of H k .
consists of those elements in T 1 moving 2, i.e., those elements in T moving both 1 and 2, and so on. Thus we have Note that Γ acts transitively on [n]. For 0 ≤ k ≤ n, from Theorem 2.6 and (5) we see that the left coset decompositions Π i (i ∈ [n]) of Γ given in (4) are equitable partitions of G k = Cay(Γ, T k ) which share the same symmetric quotient matrix In particular, B Π = B Π . To achieve our goal, we need to determine the second eigenvalue of B (8), and B (k) Π the quotient matrix of G k defined in (9). If Γ acts (k + 2)-transitively on [n], then Proof. First suppose k = 0. According to (9), we have B 11 . Similarly, for any two distinct s, t ∈ [n], there exists some σ in Γ mapping s to 1 and t to 2 by the 2-transitivity of Γ acting on [n]. Then b 12 . Combining these results, we have Thus the quotient matrix B Π . According to (9), we see that B , there is a σ ∈ Γ fixing {1, 2, . . . , k} setwise but moving s to k +1. Then σ −1 T k σ = T k and σ −1 Γ s σ = Γ k+1 by above arguments, and thus b , again by the (k + 2)-transitivity, we can choose σ ∈ Γ such that σ moves t to 2 and s to 1 but fixes {1, 2, . . . , k} setwise. Then we see that b For 1 ≤ s ≤ k and k + 1 ≤ t ≤ n, there also exists some σ in Γ mapping s to 1, t to k + 1 but fixing {1, 2, . . . , k} setwise, thus we get b Therefore, the quotient matrix B (k) Π can be written as .
where g 1 ∈ R k and g 2 ∈ R n−k are two arbitrary vectors orthogonal to the all ones vector, respectively. One can easily verify that B Π with multiplicities at least k − 1 and n − k − 1, respectively. Also note that |T k | is always an eigenvalue of B (k) Π with the all ones vector as its eigenvector because G k = Cay(Γ, T k ) is |T k |-regular. Thus there is just one eigenvalue, denoted by µ, that is not known. By computing the trace of B (k) Π in two ways, we obtain Thus the eigenvalues of B Indeed, by the (k + 2)-transitivity of Γ acting on [n], there exists some σ ∈ Γ such that σ moves 1 to k + 2 but fixes k + 1 and {2, . . . , k} setwise. Then where the last equality follows from Γ k+2 ∩ Γ k+2,k+1 = ∅. Also, we see that Combining (10) and (11) k+1,k+2 , then the result follows because b (k) k+1,k+1 ≥ 0. As above, by taking σ ∈ Γ such that σ maps 1 to k + 1 and 2 to k + 2 but fixes {3, . . . , k} setwise, we get b (k) Combining (11) and (12), we have ,k+1 | ≥ 0, and the result follows. Hence we conclude that If m < n, then we claim that G m = Cay(Γ, T m ) is disconnected. Indeed, by the definition, T m consists of those τ ∈ T such that {1, 2, . . . , m} ⊆ supp(τ ). Since each element of T has at most m supports, we have supp(τ ) = {1, 2, . . . , m} for any τ ∈ T m , which implies that T m cannot generate Γ due to m < n.
In the following, we suppose further that the action of Γ on [n] is (m + a)transitive with a ≥ 1. Under this assumption, it is clear that n ≥ m + a, and so m < n, implying that G m is disconnected. Denote by where Γ (i) is defined in (13), and T k , R k are given in (8). By definition, G k,0 = G k = Cay(Γ, T k ), H k,0 = H k = Cay(Γ k , R k ), and G k,i is the subgraph of both G k−1,i and G k,i−1 . As in Claim 3.1, the edge set of G k−1,i can be decomposed into that of G k,i and (n − i)-copies of H k,i . Also, for each fixed i, we see that is closed under conjugation in Γ (i) , and T k ∩ Γ (i) is just the set of elements in T ∩ Γ (i) moving each point of {1, 2, . . . , k} (similar as Claim 3.2). Furthermore, since n − i ≥ m + a − i ≥ m + 1, we claim that T m ⊆ Γ (i) and that G m,i = Cay(Γ (i) , T m ∩ Γ (i) ) = Cay(Γ (i) , T m ) is disconnected. In particular, we have According to Lemma 2.4 and the arguments in Section 2, every left coset decomposition of Γ (i) with respect to some stabilizer subgroup leads to an equitable partition of G k,i , and all these equitable partitions share the same quotient matrix where 0 ≤ k ≤ m − 1 and 0 ≤ i ≤ a − 1.
Before giving the main result of this section, we need the following two lemmas.
Now we give the main result of this section, which indicates that the problem of proving λ 2 (G k ) = λ 2 (B (k) Π ) (0 ≤ k ≤ m − 1) can be reduced to verifying the result for some small graphs.
Proof. If a = 1, there is nothing to prove. Thus we assume that a ≥ 2. The main idea is to prove λ 2 (G k,i ) = λ 2 (B (k,i) Π ) for all 0 ≤ k ≤ m − 1 and 0 ≤ i ≤ a − 1 by induction on k and i.
Therefore, we may conclude that According to Theorem 3.5, to prove λ 2 (G) = λ 2 (G 0 ) = λ 2 (B  ) for all k ∈ {0, 1, . . . , m − 1}. Note that if a is relatively large, i.e., the action of Γ on [n] is of high transitivity, then the graph G k,a−1 will be of small order. This makes it easier to verify the equalities. It is well known that the symmetric group S n acts n-transitively on [n], so Theorem 3.5 is particularly effective for normal Cayley graphs of S n . In the next section, we consider to determine the second eigenvalues of connected normal Cayley graphs of S n with m ≤ 5.

The second eigenvalues of normal Cayley graphs of symmetric groups
Let Γ = S n be the symmetric group on [n] with n ≥ 3. It is well known that S n acts n-transitively on [n], and that two elements in S n are conjugated if and only if they share the same cycle type. Let G = Cay(S n , T ) be a normal Cayley graph of S n , that is, T is the disjoint union of some conjugacy classes of S n . Then G is connected if and only if T contains some odd permutation. This is because T generates a nonidentity normal subgroup of S n while A n is the unique nontrivial normal subgroup of S n for n = 4, and In this section, as applications of Theorem 3.5, we consider the second eigenvalues of connected normal Cayley graphs of S n for which each element of the connection set has at most five supports.
For convenience, we first list all the nontrivial conjugacy classes of S n with each element having at most five supports: where p, q, r, s, t are pairwise distinct. For k ∈ [n], we denote by C (i) k (see Table 1) the set of elements in C (i) that moves each point of {1, 2, . . . , k}, where 1 ≤ i ≤ 6. Now suppose that G = Cay(S n , T ) (= G 0 ) is a normal Cayley graph of S n with m = max τ ∈T |supp(τ )| ≤ 5. For k ∈ [n], let T k = T \ (T ∩ (∪ k i=1 (S n ) i )) (see Claim 3.2) and G k = Cay(S n , T k ) be defined as in (8). Then T (= T 0 ) and T k (k ∈ [n]) can be respectively written as T = ∪ i∈I T C (i) (see (17)) and T k = ∪ i∈I T C (i) k (see Table 1), where I T is some nonempty subset of {1, 2, 3, 4, 5, 6}. Moreover, by the arguments at the beginning of this section, we obtain that G = Cay(S n , T ) is connected if and only if T = ∪ i∈I T C (i) with where P is the power set of {1, 2, . . . , 6}. Now we give the main result of this section, which determines the second eigenvalues of a majority of connected normal Cayley graphs (and some subgraphs of these graphs) on S n satisfying m = max τ ∈T |supp(τ )| ≤ 5.

Further research
Let Γ be finite group acts transitively on [n] (for example, Γ = S n or A n ), and let Cay(Γ, T ) be a Cayley graph of Γ. By Theorem 2.6, the left coset decomposition given in (4) is always an equitable partition of Cay(Γ, T ), and the corresponding quotient matrix B Π = (b s,t ) n×n (see (5)) is symmetric, where b s,t (=b t,s ) is the number of elements in T moving t to s. Since the eigenvalues of B Π are also eigenvalues of Cay(Γ, T ), we have λ 2 (B Π ) ≤ λ 2 (Cay(Γ, T )). Inspired by the main result of Section 4, we pose the following problem. Let T be a symmetric generating subset of Γ. We define the permutation graph Per(T ) as the edge-weighted graph with vertex set {1, 2, . . . , n} in which each edge e = st (s = t) has weight w(e) = b s,t , the number of elements in T moving t to s as mentioned above. If Γ = S n and T contains only transpositions, it is clear that the permutation graph Per(T ) coincides with the transposition graph Tra(T ) defined in Section 1. Since Cay(Γ, T ) is |T |-regular, the sum of each row of the quotient matrix B Π is equal to |T |. We can verify that B Π = |T | · I n − L(Per(T )), where L(Per(T )) is the Laplacian matrix of the permutation graph Per(T ). This implies that λ 2 (B Π ) = |T |·I n −µ n−1 (L(Per(T ))), where µ n−1 (L(Per(T ))) denotes the second least eigenvalue of L(Per(T )), i.e., the algebraic connectivity of Per(T ). Therefore, the spectral gap of Cay(Γ, T ) satisfies the inequality |T | − λ 2 (Cay(Γ, T )) ≤ |T | − λ 2 (B Π ) = µ n−1 (L(Per(T ))).
Then we can restate Problem 5.1 as below. In fact, Aldous' theorem give a positive answer of Problem 5.1 (or Problem 5.2) in the case that Γ = S n and T consists of transpositions. Also, the result of Theorem 4.1 in this paper gives a partial answer of Problem 5.1 (or Problem 5.2) for the connected normal Cayley graphs (and some of their subgraphs) of S n with max τ ∈T |supp(τ )| ≤ 5.
For any σ ∈ S n , there exists a unique partition [n] = I 1 ∪ · · · ∪ I m of [n] into contiguous blocks such that σ(I i ) = I i for each i ∈ [m]. Here, each I i consists of consecutive elements in [n], so that I i = {a, a + 1, . . . , b} for some pair of natural numbers a ≤ b. If this partition is of cardinality m, then we call σ an m-reducible permutation. In [13,14], Dai introduced and discussed some combinatorial properties of a new variant of the family of Johnson graphs, the Full-Flag Johnson graphs. He showed that the Full-Flag Johnson graph F J(n, r) (r < n) is isomorphic to the Cayley graph Cay(S n , RP (r) ), where RP (r) is the set of all (n − r)-reducible permutations of S n . For a positive integer n, the Cayley graph Cay(S n , {(i, i + 1) | 1 ≤ i ≤ n − 1}) is called the permutahedron of order n, which is a well-known combinatorial graph. Observe that each (n − 1)-reducible permutation of S n must be of the form (i, i+1) for some i ∈ [n−1], we have RP (1) and so the permutahedron of order n is just the Full-Flag Johnson graph F J(n, 1). Thus the Full-Flag Johnson graphs can be also viewed as the generalizations of permutahedra [14].
Let M n be the tridiagonal matrix of order n defined as below: At the end of the paper [14], Dai proved that the eigenvalues of M n are also eigenvalues of the permutahedron F J(n, 1), and conjectured that λ 2 (M n ) = λ 2 (F J(n, 1)). In fact, since F J(n, 1) = Cay(S n , RP (1) ) with M n is just the quotient matrix of F J(n, 1) shown in (5). Thus we may conclude that Dai's conjecture follows from Aldous' theorem immediately by the arguments at the beginning of this section. Now consider the graph F J(n, 2) = Cay(S n , RP (2) ) where RP (2) consists of all (n − 2)-reducible permutations of S n . By definition, we can check that each (n − 2)reducible permutation of S n belongs to one of the following three classes: Therefore, we have RP (2) = Q (1) ∪ Q (2) ∪ Q (3) . Furthermore, by Theorem 2.6 and (5), the graph F J(n, 2) = Cay(S n , RP (2) ) has the quotient matrix In accordance with Problem 5.1, we ask if λ 2 (F J(n, 2)) = λ 2 (B n )? Using computer, we can verify that the equality holds for 4 ≤ n ≤ 7 and we make the following conjecture.
Theorem 2.6 indicates a possible method to prove Conjecture 5.1. Now we describe the detail of the method. For k = 1, 2, we define F J k (n, 2) = Cay(S n , RP Using computer, we can check that λ 2 (F J 1 (n, 2)) = λ 2 (B n ) holds for 4 ≤ n ≤ 7, and so we propose the following conjecture.
On the other hand, for regular graphs, the smallest eigenvalue is closely related to the independent number. Let G be a k-regular graph G with smallest eigenvalue τ and independent number α(G), the well-known Hoffman ratio bound asserts that α(G) ≤ |V (G)| 1 − k/τ , and that if the equality holds for some independent set S with characteristic vector v S , then v S − |S| |V (G)| 1 is an eigenvector of the eigenvalue τ . By applying the Hoffman ratio bound to several important families of graphs belonging to classical P -or Q-polynomial association schemes (such as Johnson scheme, Hamming scheme, Grassmann scheme) and some famous Cayley graphs (such as the derangement graph) on the symmetric group S n , variants of Erdős-Ko-Rado Theorems for sets, vector spaces, integer sequences and permutations have been obtained by various researchers (see Godsil and Meagher [20] for the detail). Recently, Brouwer, Cioabȃ, Ihringer and McGinnis [6] determine the smallest eigenvalues of (distancej) Hamming graphs, (distance-j) Johnson graphs, and the graphs of the relations of classical P -and Q-polynomial association schemes. Motivated by these works, it is interesting to consider the smallest eigenvalues of normal Cayley graphs of S n . A natural question is that whether the method developed in this paper is valid for the smallest eigenvalues. However, it is not the case. According to the proof of Lemma 3.2, the quotient matrix B Π (= B 0 Π ) of the normal Cayley graph G 0 = Cay(S n , T 0 = T ) has eigenvalue |T | and |T ∩ Γ 1 | − |T ∩ Γ 2,1 | (with multiplicity n − 1). Thus we have λ n (B Π ) = λ 2 (B Π ) = |T ∩ Γ 1 | − |T ∩ Γ 2,1 |. If n ≥ 7, we can verify that λ n (B Π ) = λ 2 (B Π ) ≥ 0 holds for all connected normal Cayley graphs of S n with max τ ∈T |supp(τ )| ≤ 5, which implies that λ n (B Π ) cannot be the smallest eigenvalue. Thus we pose the following problem.