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

We study the normal Cayley graphs $\mathrm{Cay}(S_n,C(n,I\left)\right)$ on the symmetric group $S_n$, where $I\subseteq \{2,3,\dots ,n\}$ 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 $\mathrm{Cay}(S_n,C(n,I\left)\right)$ 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 $\mathrm{Cay}(S_n,C(n,I\left)\right)$ has the Aldous property, namely the strictly second largest eigenvalue is attained by the standard representation of $S_n$, and we obtain that $\mathrm{Cay}(S_n,C(n,I\left)\right)$ does not have the Aldous property whenever $n\in I$. As another corollary of our main results, we prove a recent conjecture on the second largest eigenvalue of $\mathrm{Cay}(S_n,C(n,\left\{k\right\}\left)\right)$ where $2\le k\le n-2$.