On co-edge-regular graphs with 4 distinct eigenvalues
Hong-Jun Ge, Jack H. Koolen
TL;DR
The paper refutes Tan et al.'s conjecture by constructing two infinite families of connected co-edge-regular graphs with four distinct eigenvalues and a fixed smallest eigenvalue: one family lacking $-1$ as an eigenvalue and another that is cospectral with certain clique-extensions of Latin Square graphs, thereby showing that spectral data alone cannot determine these graphs. It develops a combinatorial characterization for such graphs, leverages Haemers-Tonchev-type constructions to produce level-2 counterexamples, and introduces twisted Latin Square graphs TLS$(q,n)$ showing level-3 co-edge-regularity and cospectrality with Latin Square clique extensions. The TLS construction relies on group-divisible orthogonal arrays and parallel classes, and yields infinite families with explicit spectra, including $-1$ and another negative eigenvalue, broadening the landscape of co-edge-regular graphs with four eigenvalues. Together, these results demonstrate the richness and complexity of the spectral theory of co-edge-regular graphs and illustrate limitations of spectral characterizations in this graph class.
Abstract
Tan et al. conjectured that connected co-edge-regular graphs with four distinct eigenvalues and fixed smallest eigenvalue, when having sufficiently large valency, belong to two different families of graphs. In this paper we construct two new infinite families of connected co-edge-regular graphs with four distinct eigenvalues and fixed smallest eigenvalue, thereby disproving their conjecture. Moreover, one of these constructions demonstrates that clique-extensions of Latin Square graphs are not determined by their spectrum.