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Let $S_n$ denote the symmetric group on $n$ letters. The $k$-point fixinggraph $\mathcal{F}(n,k)$ is defined to be the graph with vertex set $S_n$ andtwo vertices $g,h$ of $\mathcal{F}(n,k)$ are joined by an edge, if and only if$gh^{-1}$ fixes exactly $k$ points. Ku, Lau and Wong [Cayley graph on symmetricgroup generated by elements fixing $k$ points, Linear Algebra Appl. 471 (2015)405-426] obtained a recursive formula for the eigenvalues of$\mathcal{F}(n,k)$.  In this paper, we use objects called excited diagrams defined as certaingeneralizations of skew shapes and derive an explicit formula for theeigenvalues of Cayley graph $\mathcal{F}(n,k)$. Then we apply this formula andshow that the eigenvalues of $\mathcal{F}(n,k)$ are in the interval$[\frac{-|S(n,k)|}{n-k-1}, |S(n,k)|]$, where $S(n,k)$ is the set of elements$\sigma$ of $S_n$ such that $\sigma$ fixes exactly $k$ points.

A hook formula for eigenvalues of k-point fixing graph