On graphs with three eigenvalues Original Research Article

We consider undirected non-regular connected graphs without loops and multiple edges (other than complete bipartite graphs) which have exactly three distinct eigenvalues (such graphs are called non-standard graphs). The interest in these graphs is motivated by the questions posed by W. Haemers during the 15th British Combinatorial Conference (Stirling, July 1995); the main question concerned the existence of such graphs. A brief review of two papers by Bridges and Mena (1979, 1981) is followed by the presentation of our new results and examples concerning, in particular, the construction of some non-standard graphs. This answers problems posed by Haemers. Other open problems are suggested and discussed in the final section.