We determine all graphs whose adjacency matrix has 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.
- Adjacency matrix
- Graph spectrum
- Spectral characterizations
Cioabă, S. M., Haemers, W. H., & Vermette, J. R. (2017). The graphs with all but two eigenvalues equal to - 2 or 0. Designs, Codes and Cryptography, 84(1-2), 153-163. https://doi.org/10.1007/s10623-016-0241-4