Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Distance-regular graphs with a few $q$-distance eigenvalues

Mamoon Abdullah, Brhane Gebremichel, Sakander Hayat, Jack H. Koolen

Abstract

In this paper we study when the $q$-distance matrix of a distance-regular graph has few distinct eigenvalues. We mainly concentrate on diameter 3.

Distance-regular graphs with a few $q$-distance eigenvalues

Abstract

In this paper we study when the -distance matrix of a distance-regular graph has few distinct eigenvalues. We mainly concentrate on diameter 3.
Paper Structure (8 sections, 15 theorems, 36 equations)

This paper contains 8 sections, 15 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Eigenvalues of a $q$-distance matrix of distance-regular graphs of diameter 3
  4. Distance-regular graphs with diameter $3$ and having $2$ distinct $\mathbb{D}_q$-eigenvalues
  5. Bipartite distance-regular graphs
  6. Antipodal distance-regular graphs
  7. Distance-regular graphs with diameter $3$ and having $3$ distinct $\mathbb{D}_q$-eigenvalues
  8. On distance eigenvalues ($q=1$) of antipodal distance-regular graphs of diameter $D$

Key Result

Lemma 2.1

Let $B$ be a real symmetric $n\times n$ matrix and $C$ a principal submatrix of $B$ of order $m$, where $m<n$. Then the eigenvalues of $C$ interlace the eigenvalues of $B$.

Theorems & Definitions (27)

  • Lemma 2.1: Cf.GD01
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • Corollary 3.5
  • Corollary 3.6
  • ...and 17 more