  Consider an experiment consisting of a set of independent trials for
comparing a set of treatments. In each trial, one treatment is chosen and the
mean response of the trial is equal to the effect of the chosen treatment. We
examine the optimal approximate designs for the estimation of a system of
treatment contrasts under such model. These approximate treatment designs can
be used to provide optimal treatment proportions for designs in more general
models with nuisance effects (e.g., time trend, effects of blocks). For any
system of pairwise treatment comparisons, we propose to represent such system
by a graph. In particular, we represent the treatment designs for these sets of
contrasts by the inverses of the vertex weights in the corresponding graph G.
We show that then the positive eigenvalues of the information matrix of a
treatment design are inverse to the positive eigenvalues of the vertex-weighted
Laplacian of G. Note that such representation of treatment designs differs from
the well known graph representation of block designs, which are represented by
edges. We provide a graph-theoretic interpretation of the D-, A- and
E-optimality for estimating sets of pairwise comparisons; as well as some
optimality results for both the systems of pairwise comparisons and the general
systems of treatment contrasts. Moreover, we provide a class of 'symmetric'
systems of treatment contrasts for which the uniform treatment design is
optimal with respect to a wide range of optimality criteria.
