The second eigenvalue of some normal Cayley graphs of high transitive groups

Let $\mathrm{\Gamma }$ be a finite group acting transitively on $[n]=\{1,2,\dots ,n\}$, and let $G=\mathrm{C}\mathrm{a}\mathrm{y}(\mathrm{\Gamma },T)$ be a Cayley graph of $\mathrm{\Gamma }$. The graph $G$ is called normal if $T$ is closed under conjugation. In this paper, we obtain an upper bound for the second (largest) eigenvalue of the adjacency matrix of the graph $G$ in terms of the second eigenvalues of certain subgraphs of $G$ (see Theorem 2.6). Using this result, we develop a recursive method to determine the second eigenvalues of certain Cayley graphs of $S_n$ and we determine the second eigenvalues of a majority of the connected normal Cayley graphs (and some of their subgraphs) of $S_n$ with $\max_{\tau\in T}|$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}$(\tau)|\leq 5$,where$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}$(\tau)$ is the set of points in $[n]$ non-fixed by $\tau$.