Solving the Ku-Wales conjecture on the eigenvalues of the derangement graph

TL;DR

This paper introduces a new recurrence formula for derangement graph eigenvalues, simplifies their sign property proof, and confirms a conjecture about eigenvalue bounds related to partition dominance order.

Contribution

It provides a novel recurrence relation for eigenvalues, simplifies the proof of their sign property, and proves a conjecture on eigenvalue bounds based on partition dominance order.

Findings

New recurrence formula for eigenvalues

Simplified proof of the Alternating Sign Property

Confirmed Ku and Wales conjecture on eigenvalue bounds

Abstract

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.