  We give a new recurrence formula for the eigenvalues of the derangement
graph. Consequently, we provide a simpler proof of the Alternating Sign
Property of the derangement graph. Moreover, we prove that the absolute value
of the eigenvalue decreases whenever the corresponding partition decreases in
the dominance order. In particular, this settles affirmatively a conjecture of
Ku and Wales (J. of Combin. Theory, Series A 117 (2010) 289--312) regarding the
lower and upper bound for the absolute values of these eigenvalues.
