Cayley graph on symmetric group generated by elements fixing k points

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