On the second largest eigenvalue of some Cayley graphs of the Symmetric Group

Let $S_n$ and $A_{n}$ denote the symmetric and alternating group on the set $\{1,.., n\},$ respectively. In this paper we are interested in the second largest eigenvalue $\lambda_{2}(\Gamma)$ of the Cayley graph $\Gamma=Cay(G,H)$ over $G=S_{n}$ or $A_{n}$ for certain connecting sets $H.$ Let $1<k\leq n$ and denote the set of all $k$-cycles in $S_{n}$ by $C(n,k).$ For $H=C(n,n)$ we prove that $\lambda_{2}(\Gamma)=(n-2)!$ (when $n$ is even) and $\lambda_{2}(\Gamma)=2(n-3)!$ (when $n$ is odd). Further, for $H=C(n,n-1)$ we have $\lambda_{2}( \Gamma)=3(n-3)(n-5)!$ (when $n$ is even) and $\lambda_{2}(\Gamma)=2(n-2)(n-5) !$ (when $n$ is odd). The case $H=C(n,3)$ has been considered in X. Huang and Q. Huang, The second largest eigenvalue of some Cayley graphs on alternating groups, J. Algebraic Combinatorics} 50(2019), $99-111$. Let $1\leq r<k<n$ and let $C(n,k;r) \subseteq C(n,k)$ be set of all $k$-cycles in $S_{n}$ which move all the points in the set $\{1,2,..., r\}.$ That is to say, $g=(i_{1},i_{2}... i_{k})(i_{k+1})\dots(i_{n})\in C(n,k;r)$ if and only if $\{1,2,..., r\}\subset \{i_{1},i_{2},..., i_{k}\}.$ Our main result concerns $\lambda_{2}( \Gamma)$, where $\Gamma=Cay(G,H)$ with $H=C(n,k;r)$ with $1\leq r<k<n$ when $G=S_{n}$ if $k$ is even and $G=A_{n}$ if $k$ is odd. Here we observe that $$\lambda_{2}( \Gamma)\geq (k-2)! {n-r \choose k-r} \frac{1}{n-r} \big((k-1)(n-k) - \frac{(k-r-1)(k-r)}{n-r-1}\big).$$ We show that this bound is sharp in the special case $k=r+1$ , giving $\lambda_{2}(\Gamma)=r!(n-r-1)$. The cases with $H=C(n,3;1)$ and $H=C(n,3;2)$ were considered earlier in the same paper of X. Huang and Q. Huang.