Alternating sign property of the perfect matching derangement graph - Koh, Samuel Zhi Kang (2023)
It was conjectured in the monograph [9] by Godsil and Meagher and in the article [10] by Lindzey that the perfect matching derangement graph M2n possesses the alternating sign property, that is, for any integer partition λ=(λ1,…,λr)⊢n, the sign of the eigenvalue ηλ of M2n is given by sign(ηλ)=(−1)n−λ1. In this paper, we prove that the conjecture is true. Our approach yields a recurrence formula for the eigenvalues of the perfect matching derangement graph as well as a new recurrence formula for the eigenvalues of the permutation derangement graph.