The second largest eigenvalue of some nonnormal Cayley graphs on symmetric groups

A Cayley graph on the symmetric group Sn is said to have the Aldous property if its strictly second largest eigenvalue (that is, the largest eigenvalue strictly smaller than the degree) is attained by the standard representation of Sn. For 1≤r<k<n, let C(n,k;r) be the set of k-cycles of Sn moving every point in {1,…,r}. Recently, Siemons and Zalesski (2022) [26] posed a conjecture which is equivalent to saying that for any n≥5 and 1≤r<k<n the nonnormal Cayley graph Cay(Sn,C(n,k;r)) on Sn with connection set C(n,k;r) has the Aldous property. Solving this conjecture, we prove that all these graphs have the Aldous property except when (i) (n,k,r)=(6,5,1) or (ii) n is odd, k=n−1, and 1≤r<n2. Along the way we determine all irreducible representations of Sn that can achieve the strictly second largest eigenvalue of Cay(Sn,C(n,n−1;r)) as well as the smallest eigenvalue of this graph.