  We investigate invariants of compact hyperk{\"a}hler manifolds introduced by
Rozansky and Witten: they associate an invariant to each graph homology class.
It is obtained by using the graph to perform contractions on a power of the
curvature tensor and then integrating the resulting scalar-valued function over
the manifold, arriving at a number. For certain graph homology classes, the
invariants we get are Chern numbers, and in fact all characteristic numbers
arise in this way.
  We use relations in graph homology to study and compare these hyperk{\"a}hler
manifold invariants. For example, we show that the norm of the Riemann
curvature can be expressed in terms of the volume and characteristic numbers of
the hyperk{\"a}hler manifold. We also investigate the question of whether the
Rozansky-Witten invariants give us something more general than characteristic
numbers. Finally, we introduce a generalization of these invariants which
incorporates holomorphic vector bundles into the construction.
