Graphs with three eigenvalues and second largest eigenvalue at most 1
Minisymposium: SPECTRAL GRAPH THEORY
Content: The classification of regular graphs with second largest eigenvalue at most 1 is equivalent to the classification of regular graphs with smallest eigenvalue at least -2. Such graphs have received a great deal of attention. Another well-studied family of graphs are regular graphs having precisely three distinct eigenvalues. These graphs are well-known to be strongly regular. The study of non-regular graphs with second largest eigenvalue at most 1 and the study of non-regular graphs with precisely three distinct eigenvalues were initiated by Hoffman and Haemers respectively. In this talk I will present some recent results about graphs having either of the properties above including a classification of graphs having both properties. This talk is based on joint work with Xi-Ming Cheng and Jack Koolen.