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

Himanshu Gupta, Allen Herman, Alice Lacaze-Masmonteil, Roghayeh Maleki, Karen Meagher

TL;DR

The paper investigates the second largest eigenvalue in graphs of the perfect matching association scheme on $K_{2n}$, with particular focus on which $S_{2n}$-irrep indices host this eigenvalue for partitions with at least two 1s. It combines trace methods, the symmetric-function framework of Srinivasan, and quotient-graph techniques to identify the $[2n-2,2]$ (and frequently $[n-1,1]$) eigenspace as carrying the second eigenvalue, proving the conjecture in the large-$n$ regime for classes of partitions and verifying it for several explicit partitions. The work also derives spectral-gap formulas, explores how gaps transform under merging parts of partitions, and provides exact results for hook partitions, linking spectral properties to combinatorial partition structure. These results have implications for mixing, expansion, and diameter properties in the perfect matching scheme and open questions about the full spectrum across all relations.

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.

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

TL;DR

The paper investigates the second largest eigenvalue in graphs of the perfect matching association scheme on , with particular focus on which -irrep indices host this eigenvalue for partitions with at least two 1s. It combines trace methods, the symmetric-function framework of Srinivasan, and quotient-graph techniques to identify the (and frequently ) eigenspace as carrying the second eigenvalue, proving the conjecture in the large- regime for classes of partitions and verifying it for several explicit partitions. The work also derives spectral-gap formulas, explores how gaps transform under merging parts of partitions, and provides exact results for hook partitions, linking spectral properties to combinatorial partition structure. These results have implications for mixing, expansion, and diameter properties in the perfect matching scheme and open questions about the full spectrum across all relations.

Abstract

The perfect matching association scheme is a set of relations on the perfect matchings of the complete graph on 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 indexed by the even partitions of . 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 . We confirm this conjecture for matrices labeled by the partitions , and , as well as any partition in which the first part is sufficiently large.
Paper Structure (9 sections, 25 theorems, 127 equations, 4 figures, 8 tables)

This paper contains 9 sections, 25 theorems, 127 equations, 4 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Combinatorial formulae for the spectrum of A2n
  4. Quotient graphs
  5. Proof of Conjecture \ref{['conj:main']} for n sufficiently large
  6. Proof of Conjecture \ref{['conj:main']} for certain matrices using symmetric functions
  7. Relationship between conjectured spectral gaps of certain matrices
  8. Future Work
  9. Tables of Eigenvalues of perfect matching association scheme for small n

Key Result

Lemma 2.1

MacDonald Let $\mu = [\mu_1^{m_1}, \dots, \mu_k^{m_k}]$ and $n=\sum_{i=1}^k m_i \mu_i$. Then the valency of $A_\mu$ is given by

Figures (4)

  • Figure 1: Two perfect matchings of $K_{12}$ drawn in dashed and solid, respectively, that overlap to form three cycles of lengths $6, 4$, and $2$, respectively.
  • Figure 2: Young tableau for partition $2\lambda=[6,4,2]$ with content.
  • Figure 3: Illustrating an admissible construction of $\lambda^+$ from $\lambda$.
  • Figure 4: Illustration of the merging of a $4$-cyle with a $6$-cycle by the action of $\sigma=(a,d)$ on perfect matching $Q$.

Theorems & Definitions (45)

  • Definition 1.1
  • Definition 1.2
  • Conjecture 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Corollary 2.6
  • Lemma 2.7
  • ...and 35 more