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

TL;DR

This paper investigates the structure of regular graphs with specific spectral properties, characterizing those with a small dual diameter in their spectral idempotent algebra, and classifying related distance-regular graphs and association schemes.

Contribution

It provides a combinatorial characterization of graphs with a spectral idempotent algebra of dimension two, including classifications of distance-regular graphs and graphs with multiple eigenvalues.

Findings

Classified distance-regular graphs with the property

Identified graphs generating 3-class association schemes

Determined all graphs with four eigenvalues and two eigenvalues of a certain spectral property

Abstract

Let $\Gamma$ be a connected regular graph with an eigenvalue $\lambda$ and corresponding idempotent $E_{\lambda}$. Let ${\cal E}_{\lambda}=\langle J,E_{\lambda}\rangle^\circ$ be the algebra generated by $J$ and $E_\lambda$ with respect to the entrywise-Hadamard product, where $J$ is the all-$1$ matrix. We study the combinatorial structure of a graph $\Gamma$ for which ${\cal E}_{\lambda}$ 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 $\lambda$ for which ${\rm dim}({\cal E}_{\lambda})=2$ and determine all such graphs with four distinct eigenvalues.