The second largest eigenvalue of normal Cayley graphs on symmetric groups generated by cycles

Abstract

We study the normal Cayley graphs Cay(S n , C(n, I)) on the symmetric group S n , where I ⊆ {2, 3, . . . , n} and C(n, I) is the set of all cycles in S n with length in I. We prove that the strictly second largest eigenvalue of Cay(S n , C(n, I)) can only be achieved by at most four irreducible representations of S n , and we determine further the multiplicity of this eigenvalue in several special cases. As a corollary, in the case when I contains neither n − 1 nor n we know exactly when Cay(S n , C(n, I)) has the Aldous property, namely the strictly second largest eigenvalue is attained by the standard representation of S n , and we obtain that Cay(S n , C(n, I)) does not have the Aldous property whenever n ∈ I. As another corollary of our main results, we prove a recent conjecture on the second largest eigenvalue of Cay(S n , C(n, {k})) where 2 ≤ k ≤ n − 2.