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.