Let be the symmetric group on n-points. The k-point fixing graph is defined to be the graph with vertex set and two vertices g, h of are joined if and only if fixes exactly k points. In this paper, we give a recurrence formula for the eigenvalues of a class of regular subgraphs of . By using this recurrence formula, we will determine the smallest eigenvalues for this class of regular subgraphs of for sufficiently large n.