The graphs with all but two eigenvalues equal to pm 1

摘要

We determine all graphs whose adjacency matrices have at most two eigenvalues (multiplicities included) different from (pm 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.