The graphs with all but two eigenvalues equal to +/- 1

摘要

We determine all graphs whose adjacency matrices have at most two eigenvalues (multiplicities included) different from +/- 1 and decide which of these graphs are determined by their spectrum. This includes the so-called friendship graphs, which consist of a number of edge-disjoint triangles meeting in one vertex. It turns out that the friendship graph is determined by its spectrum, except when the number of triangles equals sixteen.