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The coherent rank of a graph with three eigenvalues

Gary Greaves, Jose Yip

Abstract

We characterise graphs that have three distinct eigenvalues and coherent ranks 8 and 9, linking the former to certain symmetric 2-designs and the latter to specific quasi-symmetric 2-designs. This characterisation leads to the discovery of a new biregular graph with three distinct eigenvalues. Additionally, we demonstrate that the coherent rank of a triregular graph with three distinct eigenvalues is at least 14. Finally, we introduce a conjecturally infinite family of biregular graphs with three distinct eigenvalues, obtained by switching the block graphs of orthogonal arrays.

The coherent rank of a graph with three eigenvalues

Abstract

We characterise graphs that have three distinct eigenvalues and coherent ranks 8 and 9, linking the former to certain symmetric 2-designs and the latter to specific quasi-symmetric 2-designs. This characterisation leads to the discovery of a new biregular graph with three distinct eigenvalues. Additionally, we demonstrate that the coherent rank of a triregular graph with three distinct eigenvalues is at least 14. Finally, we introduce a conjecturally infinite family of biregular graphs with three distinct eigenvalues, obtained by switching the block graphs of orthogonal arrays.

Paper Structure

This paper contains 27 sections, 46 theorems, 104 equations, 8 tables.

Table of Contents

  1. Introduction
  2. Graphs with three distinct eigenvalues
  3. Fundamentals
  4. Cones and bipartite graphs
  5. Valencies and equitable partitions
  6. Coherent configurations and algebras
  7. The coherent rank
  8. Small coherent rank
  9. Coherent rank 8
  10. Symmetric 2-designs
  11. Graphs with three eigenvalues and coherent rank 8
  12. Coherent rank 9
  13. The total graph and whole graph of a QSD
  14. Underlying quasi-symmetric designs
  15. Parametric quasi-symmetric designs
  16. ...and 12 more sections

Key Result

Theorem \oldthetheorem

Let $A$ be a real symmetric $n \times n$ matrix, and $B$ be an $m \times m$ principal submatrix of $A$. Suppose $A$ has eigenvalues $\lambda_1 \leqslant \dots \leqslant \lambda_n$, and $B$ has eigenvalues $\mu_1 \leqslant \dots \leqslant \mu_m$. Then

Theorems & Definitions (80)

  • Theorem \oldthetheorem: Eigenvalue Interlacing Theorem
  • Theorem \oldthetheorem: Cheng_2016
  • Lemma \oldthetheorem: VANDAM1998101
  • Lemma \oldthetheorem
  • Theorem \oldthetheorem: Cheng_2016
  • Lemma \oldthetheorem: Brouwer:SpectraGraphs
  • Theorem \oldthetheorem: VANDAM1998101
  • Theorem \oldthetheorem: Cheng_2016
  • Lemma \oldthetheorem
  • Corollary \oldthetheorem
  • ...and 70 more