On the combinatorial structure of graphs with a spectral idempotent of small dual diameter
Combinatorics
2026-03-25 v1
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.
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}
}