Weak Field Expansion of Gravity and Graphical Representation

We introduce a graphical representation for a global SO(n) tensor $\pl_\m\pl_\n h_\ab$, which generally appears in the perturbative approach of gravity around the flat space: $g_\mn=\del_\mn+h_\mn$. We systematically construct global SO(n) invariants. Independence and completeness of those invariants are shown by taking examples of $\pl\pl h$-, and $ (\pl\pl h)^2$- invariants. They are classified graphically. Indices which characterize all independent invariants (or graphs) are given. We apply the results to general invariants with dimension $(Mass)^4$ and the Gauss-Bonnet identity in 4-dim gravity.