On the second largest eigenvalue of certain graphs in the perfect matching association scheme

TL;DR

This paper investigates the eigenvalues of matrices in the perfect matching association scheme, conjecturing the second largest eigenvalue's eigenspace and confirming it for specific partitions and cases.

Contribution

It proposes a conjecture about the eigenspace of the second largest eigenvalue in the perfect matching association scheme and verifies it for multiple partition types.

Findings

Conjecture about the second largest eigenvalue's eigenspace.

Verification for specific partitions including [2, 1^{n-2}] and others.

Confirmation for partitions with a sufficiently large first part.

Abstract

The perfect matching association scheme is a set of relations on the perfect matchings of the complete graph on $2n$ vertices. The relations between perfect matchings are defined by the cycle structure of the union of any two perfect matchings, and each relation can be represented as a matrix. Each matrix is labeled by an integer partition whose parts correspond to the size do the cycles in the union. Since these matrices form an association scheme, they are simultaneously diagonalizable. Further, it is well-known that the common eigenspaces correspond to the irreducible representations of $S_{2n}$ indexed by the even partitions of $2n$. In this paper, we conjecture that the second largest eigenvalue of the matrices in the perfect matching association scheme labeled by a partition containing at least two parts of size 1 always occurs on the eigenspace corresponding to the representation indexed by $[2n-2, 2]$. We confirm this conjecture for matrices labeled by the partitions $[2, 1^{n-2}], [3, 1^{n-3}], [2, 2, 1^{n-4}], [4, 1^{n-4}], [3, 2, 1^{n-5}]$, and $[5, 1^{n-5}]$, as well as any partition in which the first part is sufficiently large.