Graphs having three distinct eigenvalues are a fundamental object of study in spectral graph theory. Strongly regular graphs are the most well-studied examples. In 1995, at the 15th British Combinatorial Conference, Willem Haemers asked do there exist any connected graphs having three distinct eigenvalues apart from strongly regular graphs and complete bipartite graphs. Haemer’s question prompted responses from Muzychuk-Klin and Van Dam who found new families of nonregular graphs having three distinct eigenvalues.
Muzychuk and Klin initiated the study of a graph with three distinct eigenvalues via its Weisfeiler-Leman closure. They classified such graphs whose Weisfeiler-Leman closure has rank at most 7. In this talk, we extend this classification up to rank 9. Our results include a correction of the literature (where the rank 8 case was erroneously claimed to be impossible) and discussion of further study.