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1. 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… (More)

We review the construction of integral Apollonian circle packings. There are a number of Diophantine problems that arise in the context of such packings. We discuss some of them and describe some recent advances. 1. AN INTEGRAL PACKING. The quarter, nickel, and dime in Figure 1 are placed so that they are mutually tangent. This configuration is unique up to… (More)

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