Cayley Graph on Symmetric Group Generated by Elements Fixing $k$ Points

Let $\mathcal{S}_{n}$ be the symmetric group on $[n]=\{1, \ldots, n\}$. The $k$-point fixing graph $\mathcal{F}(n,k)$ is defined to be the graph with vertex set $\mathcal{S}_{n}$ and two vertices $g$, $h$ of $\mathcal{F}(n,k)$ are joined if and only if $gh^{-1}$ fixes exactly $k$ points. In this paper, we derive a recurrence formula for the eigenvalues of $\mathcal{F}(n,k)$. Then we apply our result to determine the sign of the eigenvalues of $\mathcal{F}(n,1)$.