Studying the graphs with few distinct adjacency eigenvalue was first raised by results about graph with few distinct Laplacia eigenvalue were obtained by Haemer and van Dam in 1995. Graphs with one (resp. two) distinct eigenvalues are trivial (resp. complete) graphs. The well known family of graphs with three distinct eigenvalues is strongly regular graphs. In fact, each regular graph with three distinct eigenvalues is strongly regular. Moreover, each complete bipartite graph has three distinct adjacency eigenvalues. The problem Doob in 1970. Some of characterization of non-regular, non-bipartite graphs with three distinct adjacency eigenvalues was raised by Haemers in 1995. In this thesis we summarize the well known results o graphs with three distinct adjacency and Laplacian eigenvalues and we give some new facts on those with three distinct signless Laplacian eigenvalues. In the sequel, we investigate the mention problems for integral graphs.
