Background.The aim of this note is to give an explicit construction of a rich family of k-regular (except for k ° =k) of the adjacency matrix satisfy Ikjl < 2 k~-l.graphs for which all the eigenvalues kj This bound is optimal (see Proposition 2.1).We call such graphs Ramanujan graphs.These graphs have many applications in the construction of explicit algorithms.~ny of these applications come from the fact that Ramanujan graphs make good expanders.Expanders in ~irn serve as basic building blocks for the construction of nonblocking networks [Pin, Bas-Pin, PI1], s~perconcentrators [G-G], sorting and selecting algorithms [AKS, PI3] etc.More precisely let X = (V,E) be a graph with vertices V and edges E .For A c: V we define the boundary of A, denoted ~A, by 8A = [yc V[d(y,A) = i} .