PhDA typical problem in experimental design theory is to find a block
design in a class that is optimal with respect to some criteria, which
are usually convex functions of the Laplacian eigenvalues. Although this
question has a statistical background, there are overlaps with graph and
design theory: some of the optimality criteria correspond to graph properties
and designs considered ‘nice’ by combinatorialists are often optimal.
In this thesis we investigate this connection from a combinatorial point
of view.
We extend a result on optimality of some generalized polygons, in
particular the generalized hexagon and octagon, to a third optimality criterion.
The E-criterion is equivalent with the graph theoretical problem
of maximizing the algebraic connectivity. We give a new upper bound for
regular graphs and characterize a class of E-optimal regular graph designs
(RGDs). We then study generalized hexagons as block designs and
prove some properties of the eigenvalues of the designs in that class. Proceeding
to higher-dimensional geometries, we look at projective spaces
and find optimal designs among two-dimensional substructures. Some
new properties of Grassmann graphs are proved. Stepping away from
the background of geometries, we study graphs obtained from optimal
graphs by deleting one or several edges. This chapter highlights the currently
available methods to compare graphs on the A- and D-criteria.
The last chapter is devoted to designs to which a number of blocks are
added. Cheng showed that RGDs are A- and D-optimal if the number of
blocks is large enough for which we give a bound and characterize the best
RGDs in terms of their underlying graphs. We then present the results
of an exhaustive computer search for optimal RGDs for up to 18 points.
The search produced examples supporting several open conjectures
