RAMANUJAN DIAGRAMS

A diagram D is a graph that is of finite volume with respect to a meas ure (weights) on the vertices and edges. The author gives the basic de finitions for a diagram, and defines the cases where it is an expander . Let Delta be the Laplacian on L(2)(D), and let lambda be the infimum of its spectrum on the subspace of functions that are orthogonal to t he constant function. The strong connection between lambda being large and D being a good expander is shown. For a k-regular infinite diagra m, the largest possible lambda is k - 2 root k-1, and when this is ach ieved, it is called a Ramanujan diagram. Using representation theory o f PGL(2), many infinite families of infinite Ramanujan diagrams are ex plicitly constructed.