Key points are not available for this paper at this time.
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.
Lubotzky et al. (Wed,) studied this question.