The smallest eigenvalues of the 1-point fixing graph

Let Sn be the symmetric group on {1,…,n}. The k-point fixing graph F(n,k) is defined to be the graph with vertex set Sn and two vertices g, h of F(n,k) are joined if and only if gh−1 fixes exactly k points. In this paper, we determine the smallest eigenvalue for F(n,1).