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Aldous property for full-flag Johnson graphs

Gary Greaves, Haoran Zhu

Abstract

We show that the full-flag Johnson graph has spectral gap equal to that of its Schreier quotient arising from the point-stabiliser equitable partition. Our results confirm two conjectures posed by Huang, Huang, and Cioabă, which imply an Aldous-type spectral-gap phenomenon for full-flag Johnson graphs.

Aldous property for full-flag Johnson graphs

Abstract

We show that the full-flag Johnson graph has spectral gap equal to that of its Schreier quotient arising from the point-stabiliser equitable partition. Our results confirm two conjectures posed by Huang, Huang, and Cioabă, which imply an Aldous-type spectral-gap phenomenon for full-flag Johnson graphs.
Paper Structure (8 sections, 15 theorems, 70 equations, 1 figure)

This paper contains 8 sections, 15 theorems, 70 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Cayley graphs and equitable partitions
  3. Full-flag Johnson graph as a Cayley graph
  4. Schreier graphs and quotient matrices
  5. Overview of the proof
  6. Two eigenvalue inequalities
  7. The first inequality
  8. The second inequality

Key Result

Theorem 1.1

For each $n\ge 4$, the full-flag Johnson graph ${{F\!\!J}}(n,2)$ has the Aldous property.

Figures (1)

  • Figure 1: The graph $F\!\!J(4,2)$ with vertices labelled by the image of 1234 under the action of the corresponding permutation. The vertices are grouped with respect to the point-stabiliser partition $\Pi_4$.

Theorems & Definitions (24)

  • Theorem 1.1
  • Lemma 2.1: Godsil and Royle GR, Theorem 9.3.3
  • Lemma 2.2: HHC
  • Remark 2.3
  • Theorem 2.4: HHC
  • Lemma 2.5: HHC
  • Lemma 2.6: HHC
  • proof
  • Lemma 2.7: HHC
  • proof
  • ...and 14 more