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

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.