  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
$\mathcal{F}(n,1)$.
