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

We study the normal Cayley graphs $\mathrm{C}\mathrm{a}\mathrm{y}(S_n,C(n,I))$ 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 $\mathrm{C}\mathrm{a}\mathrm{y}(S_n,C(n,I))$ 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 $\mathrm{C}\mathrm{a}\mathrm{y}(S_n,C(n,I))$ has the Aldous property, namely the strictly second largest eigenvalue is attained by the standard representation of $S_n$, and we obtain that $\mathrm{C}\mathrm{a}\mathrm{y}(S_n,C(n,I))$ 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 $\mathrm{C}\mathrm{a}\mathrm{y}(S_n,C(n,\{k\}))$ where $2\le k\le n-2$.