Eigenvalues of graphs, non-regular graphs, few eigenvalues, strong graphs, regular two-graphs, Seidel switching.
Dom de Caen posed the question whether connected graphs with three distinct eigenvalues have at most three distinct valencies. We do not answer this question, but instead construct connected graphs with four and five distinct eigenvalues and arbitrarily many distinct valencies. The graphs with four distinct eigenvalues come from regular two-graphs. As a side result, we characterize the disconnected graphs and the graphs with three distinct eigenvalues in the switching class of a regular two-graph.
Dam, Edwin van; Koolen, Jack H.; and Xia, Zheng-Jiang.
"Graphs with many valencies and few eigenvalues",
Electronic Journal of Linear Algebra,