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Some integer values in the spectra of burnt pancake graphs

Saúl A. Blanco, Charles Buehrle

Abstract

The burnt pancake graph, denoted by $\mathbb{BP}_n$, is formed by connecting signed permutations via prefix reversals. Here, we discuss some spectral properties of $\mathbb{BP}_n$. More precisely, we prove that the adjacency spectrum of $\mathbb{BP}_n$ contains all integer values in the set $\{0, 1, \ldots, n\}\setminus\{\left\lfloor n/2 \right\rfloor\}$.

Some integer values in the spectra of burnt pancake graphs

Abstract

The burnt pancake graph, denoted by , is formed by connecting signed permutations via prefix reversals. Here, we discuss some spectral properties of . More precisely, we prove that the adjacency spectrum of contains all integer values in the set .
Paper Structure (4 sections, 7 theorems, 11 equations, 1 figure)

This paper contains 4 sections, 7 theorems, 11 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Regular partitions
  3. Some integer eigenvalues
  4. Final remarks

Key Result

Proposition 1

( Dalfo) The spectrum of $\mathbb{P}_n$ with $n\geq3$ contains every element in the set $[-1,n-1]\setminus\{\lfloor (n-2)/2 \rfloor\}$.

Figures (1)

  • Figure 1: The graph $\tilde{B}$, a projection of $\mathbb{BP}_n$

Theorems & Definitions (12)

  • Proposition 1
  • Lemma 2: Lemma 1.1 in Dalfo
  • Proposition 3: Proposition 2.1 in Dalfo
  • Lemma 4
  • proof
  • Theorem 5
  • proof
  • Definition 3.1
  • Corollary 6
  • Proposition 7
  • ...and 2 more