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 ${\mu }^{\prime }$ of a positive integer n with the same first part u and the lexicographic order $\mu ◃{\mu }^{\prime }$, the eigenvalues ${\xi }_{\mu }$ and ${\xi }_{{\mu }^{\prime }}$ of the derangement graph ${\Gamma }_n$ have the property $\left|{\xi }_{\mu }\right|\le \left|{\xi }_{{\mu }^{\prime }}\right|$, 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 $M_{2n}$ of $K_{2n}$ by finding a new recurrence formula for the eigenvalues of $M_{2n}$.