The absolute values of the perfect matching derangement graph’s eigenvalues almost follow the lexicographic order of partitions

In 2013, Ku and Wong showed that for any partitions μ and μ' of a positive integer n with the same first part u and the lexicographic order μ ⊲ μ', the eigenvalues ξ_μ and ξ_μ' of the derangement graph Γ_η have the property | ξ_μ| ≤ |ξ_μ'|, where the equality holds if and only if u = 3 and all other parts are less than 3. In this article, we obtain an analogous conclusion on the eigenvalues of the perfect matching derangement graph
of M_2n of K_2n by finding a new recurrence formula for the eigenvalues of M_2n.