English

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

Combinatorics 2026-03-25 v1

Abstract

Let Γ\Gamma be a connected regular graph with an eigenvalue λ\lambda and corresponding idempotent EλE_{\lambda}. Let Eλ=J,Eλ{\cal E}_{\lambda}=\langle J,E_{\lambda}\rangle^\circ be the algebra generated by JJ and EλE_\lambda with respect to the entrywise-Hadamard product, where JJ is the all-11 matrix. We study the combinatorial structure of a graph Γ\Gamma for which Eλ{\cal E}_{\lambda} has dimension 22, 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 33-class association scheme. We also study the graphs that have two eigenvalues λ\lambda for which dim(Eλ)=2{\rm dim}({\cal E}_{\lambda})=2 and determine all such graphs with four distinct eigenvalues.

Keywords

Cite

@article{arxiv.2603.22601,
  title  = {On the combinatorial structure of graphs with a spectral idempotent of small dual diameter},
  author = {Edwin R. van Dam and Giusy Monzillo and Safet Penjić},
  journal= {arXiv preprint arXiv:2603.22601},
  year   = {2026}
}