Three-eigenvalue characterization of a connected strongly regular graph
= Three-eigenvalue characterization of a connected strongly regular graph
A connected regular graph is strongly regular exactly when its adjacency matrix has three distinct eigenvalues, apart from the complete graphs, which have two. The forward direction follows from the <adjacency-matrix relation for a strongly regular graph>; conversely, the quadratic polynomial vanishing on the two nonconstant eigenspaces is a scalar multiple of the <all-ones matrix>.