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A note on some spectral properties of generalised pancake graphs

Gary R. W. Greaves, Haoran Zhu

Abstract

We prove that the spectral gap of generalised pancake graphs is strictly less than 2 and strictly less than 1 for burnt pancake graphs. In addition, we establish lower bounds on the multiplicities of certain integer eigenvalues of generalised pancake graphs. Together, these results settle two recent conjectures of Blanco and Buehrle.

A note on some spectral properties of generalised pancake graphs

Abstract

We prove that the spectral gap of generalised pancake graphs is strictly less than 2 and strictly less than 1 for burnt pancake graphs. In addition, we establish lower bounds on the multiplicities of certain integer eigenvalues of generalised pancake graphs. Together, these results settle two recent conjectures of Blanco and Buehrle.

Paper Structure

This paper contains 5 sections, 7 theorems, 23 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Some spectral properties of $\mathcal{P}_m(n)$
  3. Equitable partitions and interlacing
  4. The equitable partition $\mathfrak P_{m,n}$
  5. Proofs of the main results

Key Result

Theorem 1.1

Let $n, m \geqslant 2$ be integers. Then

Figures (1)

  • Figure 1: Generalised pancake graph $\mathcal{P}_3(2)$ with its vertices grouped according to the vertex partition $\mathfrak P_{3,2}$.

Theorems & Definitions (12)

  • Theorem 1.1: Blanco-Buehrle's spectral gap conjecture
  • Theorem 1.2: Blanco-Buehrle's multiplicity conjecture
  • Lemma 2.1: F
  • Lemma 2.2: Godsil and Royle GR, Theorem 9.3.3
  • Lemma 2.3
  • proof
  • Lemma 2.4: Friedman
  • Lemma 2.5
  • proof : Proof of Theorem \ref{['thm:main']}
  • proof : Proof of Theorem \ref{['thm:mult']}
  • ...and 2 more