Abstract

Graphs with few distinct eigenvalues form an interesting class of graphs. Clearly if all the eigenvalues of a graph coincide, then we have a trivial graph (a graph without edges). Connected graphs with only two distinct eigenvalues are proven to be complete graphs. The first non-trivial graphs with three distinct eigenvalues are the strongly regular graphs. Graphs with exactly three distinct eigenvalues are generalizations of strongly regular graphs by dropping regularity. A large family of (in general) non-regular examples is given by the complete bipartite graphs $K_{m,n}$. Other examples were found by Bridges, Mena, Muzychuk and Klin, most of them being cones. Those with the least eigenvalue $-2$ have been characterized by Van Dam. In this talk we give some results on graphs with few distinct eigenvalues. Moreover we consider graphs with three distinct eigenvalues and we characterize those with the largest eigenvalue less than 8.



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