  A statistician designing an experiment wants to get as much information as
possible from the data gathered. Often this means the most precise estimate
possible (that is, an estimate with minimum possible variance) of the unknown
parameters. If there are several parameters, this can be interpreted in many
ways: do we want to minimize the average variance, or the maximum variance, or
the volume of a confidence region for the parameters?
  In the case of block designs, these optimality criteria can be calculated
from the concurrence graph of the design, and in many cases from its Laplacian
eigenvalues. The Levi graph can also be used. The various criteria turn out to
be closely connected with other properties of the graph as a network, such as
number of spanning trees, isoperimetric number, and the sum of the resistances
between pairs of vertices when the graph is regarded as an electrical network.
  In this chapter, we discuss the notions of optimality for incomplete-block
designs, explain the graph-theoretic connections, and prove some old and new
results about optimality.
