Solving the Ku-Wales conjecture on the eigenvalues of the derangement   graph
  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. 
Cheng Yeaw Ku and Kok Bin Wong
math.CO