The smallest eigenvalues of the 1-point fixing graph

Eigenvalues of the matching derangement graph

“…. , λ r ) n, r ≥ 2 and λ = (n − 1, 1) or (2, 1 n−2 ), then the sign of 1 ξ λ is equal to (−1) n−λ 1 −1 (see also [2,3,[10][11][12][13] for other related results). It was conjectured in [6,7] that the alternating sign property holds for the matching derangement graph.…”

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

“…In particular, D n := T (n, {n}) is the set of derangements on [n] and F (n, 0) = Cay(S n , D n ) is widely known as the derangement graph on [n]. Theorem 1.4 together with some known results from [9,26,34] implies the following corollary, which asserts that, for sufficiently large n, F (n, 0) and F (n, 1) are the only graphs among F (n, k) (0 ≤ k ≤ n − 2) without the Aldous property.…”

“…Note that, for 0 ≤ k ≤ n − 2, T (n, {n − k}) is the set of permutations of S n fixing exactly k points, and the normal Cayley graph Cay(S n , T (n, {n − k})) is exactly the k-point-fixing graph F (n, k) studied in [26]. In particular, D n := T (n, {n}) is the set of derangements on [n] and F (n, 0) = Cay(S n , D n ) is widely known as the derangement graph on [n].…”

“…In [26], Ku, Lau and Wong proved that for n ≥ 7 the smallest eigenvalue of F (n, 1) is achieved only by the irreducible representation of S n corresponding to the partition (n − 2, 2). In [26,Lemma 3.5], they also proved that for n 7 the standard representation of S n yields the eigenvalue 0 of F (n, 1) while at least one of the partitions (1 n ), (2 2 , 1 n−4 ), (3, 1 n−3 ) produces a positive eigenvalue of F (n, 1) other than…”

The spectrum of eigenvalues for certain subgraphs of the k-point fixing graph

“…Eigenvalues of F(n, k) was first studied in [14]. The smallest eigenvalue of F(n, 1) was determined in [16]. The partition associated to the smallest eigenvalue of F(n, k) was determined in [17].…”