  In this article we develop a graphical calculus for stable invariants of
Riemannian manifolds akin to the graphical calculus for Rozansky-Witten
invariants for hyperk\"ahler manifolds; based on interpreting trivalent graphs
with colored edges as stably invariant polynomials on the space of algebraic
curvature tensors. In this graphical calculus we describe explicitly the
Pfaffian polynomials central to the Theorem of Chern-Gau{\ss}-Bonnet and the
normalized moment polynomials calculating the moments of sectional curvature
considered as a random variable on the Gra{\ss}mannian of planes. Eventually we
illustrate the power of this graphical calculus by deriving a curvature
identity for compact Einstein manifolds of dimensions greater than 2 involving
the Euler characteristic, the third moment of sectional curvature and the
$L^2$--norm of the covariant derivative of the curvature tensor. A model
implementation of this calculus for the computer algebra system Maxima is
available for download under http://www.matcuer.unam.mx/~gw/CurvGraphs.mac.
