Eigenvalues of the Derangement Graph

TL;DR

This paper studies the eigenvalues of the derangement graph on the symmetric group, revealing how their signs relate to partition shapes and providing bounds on their magnitudes.

Contribution

It characterizes the sign of eigenvalues based on Ferrers diagram shape and offers bounds for their absolute values, advancing understanding of the graph's spectral properties.

Findings

Eigenvalue sign depends on the parity of cells below the first row.

Bounds established for eigenvalue magnitudes.

Eigenvalues indexed by partitions of n.

Abstract

We consider the Cayley graph on the symmetric group Sn generated by derangements. It is well known that the eigenvalues of this grpah are indexed by partitions of n. We investigate how these eigenvalues are determined by the shape of their corresponding partitions. In particular, we show that the sign of an eigenvalue is the parity of the number of cells below the first row of the corresponding Ferrers diagram. We also provide some lower and upper bounds for the absolute values of these eigenvalues.