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On the combinatorial structure of graphs with a spectral idempotent of small dual diameter

Edwin R. van Dam, Giusy Monzillo, Safet Penjić

Abstract

Let $Γ$ be a connected regular graph with an eigenvalue $λ$ and corresponding idempotent $E_λ$. Let ${\cal E}_λ=\langle J,E_λ\rangle^\circ$ be the algebra generated by $J$ and $E_λ$ with respect to the entrywise-Hadamard product, where $J$ is the all-$1$ matrix. We study the combinatorial structure of a graph $Γ$ for which ${\cal E}_λ$ has dimension $2$, giving a combinatorial characterization of such graphs in terms of equitable partitions. We present many examples and classify the distance-regular graphs with this property, as well as graphs that generate a $3$-class association scheme. We also study the graphs that have two eigenvalues $λ$ for which ${\rm dim}({\cal E}_λ)=2$ and determine all such graphs with four distinct eigenvalues.

On the combinatorial structure of graphs with a spectral idempotent of small dual diameter

Abstract

Let be a connected regular graph with an eigenvalue and corresponding idempotent . Let be the algebra generated by and with respect to the entrywise-Hadamard product, where is the all- matrix. We study the combinatorial structure of a graph for which has dimension , giving a combinatorial characterization of such graphs in terms of equitable partitions. We present many examples and classify the distance-regular graphs with this property, as well as graphs that generate a -class association scheme. We also study the graphs that have two eigenvalues for which and determine all such graphs with four distinct eigenvalues.
Paper Structure (9 sections, 3 theorems, 2 equations)

This paper contains 9 sections, 3 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graphs and matrices
  4. Some properties of the quotient matrix of an equitable partition
  5. Association schemes and distance-regular graphs
  6. Imprimitivity
  7. Distance-regular graphs
  8. The indubitable partition
  9. Idempotents in the adjacency algebra

Key Result

Theorem 1.2

Let $\Gamma$ be a connected regular graph on $v$ vertices that has an eigenvalue $\lambda$ with multiplicity $m$ and corresponding spectral idempotent $E_{\lambda}$. Let ${\cal E}={\cal E}_{\lambda}=\langle J,E_{\lambda}\rangle^\circ$. Then ${\cal E}$ has dimension $2$ if and only if $\Gamma$ has a

Theorems & Definitions (4)

  • Definition 1.1
  • Theorem 1.2
  • Lemma 2.1: GR
  • Lemma 3.1