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

We study the normal Cayley graphs Cay (Sn, C (n, I) ) on the symmetric group Sn, where I⊆2, 3, …, n and C (n, I) is the set of all cycles in Sn with length in I. We prove that the strictly second largest eigenvalue of Cay (Sn, C (n, I) ) can only be achieved by at most four irreducible representations of Sn, 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 (Sn, C (n, I) ) has the Aldous property, namely the strictly second largest eigenvalue is attained by the standard representation of Sn, and we obtain that Cay (Sn, 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 (Sn, C (n, k) ) where 2≤k≤n−2.