  We present some approaches to the perturbative analysis of the classical and
quantum gravity. First we introduce a graphical representation for a global
SO(n) tensor $(\pl)^d h_\ab$, which generally appears in the weak field
expansion around the flat space: $g_\mn=\del_\mn+h_\mn$. Making use of this
representation, we explain 1) Generating function of graphs (Feynman diagram
approach), 2) Adjacency matrix (Matrix approach), 3) Graphical classification
in terms of "topology indices" (Topology approach), 4) The Young tableau
(Symmetric group approach). We systematically construct the global SO(n)
invariants. How to show the independence and completeness of those invariants
is the main theme. We explain it taking simple examples of $\pl\pl h-, {and}
(\pl\pl h)^2-$ invariants in the text. The results are applied to the analysis
of the independence of general invariants and (the leading order of) the Weyl
anomalies of scalar-gravity theories in "diverse" dimensions (2,4,6,8,10
dimensions).
