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

A Cayley graph on the symmetric group $S_n$ 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 $S_n$. For $1\le r<k<n$, let $C(n,k;r)$ be the set of $k$-cycles of $S_n$ moving every point in $\{1,\dots ,r\}$. Recently, Siemons and Zalesski [J. Algebraic Combin. 55 (2022) 989–1005] posed a conjecture which is equivalent to saying that for any $n\ge 5$ and $1\le r<k<n$ the nonnormal Cayley graph $\mathrm{C}\mathrm{a}\mathrm{y}(S_n,C(n,k;r))$ on $S_n$ 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 $1\le r<\frac{n}{2}$. Along the way we determine all irreducible representations of $S_n$ that can achieve the strictly second largest eigenvalue of $\mathrm{C}\mathrm{a}\mathrm{y}(S_n,C(n,n-1;r))$ as well as the smallest eigenvalue of this graph.