  The ring of graph invariants is spanned by the basic graph invariants which
calculate the number of subgraphs isomorphic to a given graph in other graphs.
These subgraphs counting invariants are not algebraically independent. In our
view the most important problem in graph theory of unlabeled graphs is the
problem of determining graphic values of arbitrary sets of graph invariants.
This corresponds to explaining the syzygy of the graph invariants when the
number of vertices is unbounded. We introduce two methods to explore this
complicated structure. Sets of graphs with a small number of vertices impose
constraints on larger sets. We describe families of inequalities of graph
invariants. These inequalities allow to loop over all values of graph
invariants which look like graphic from the small sets point of view. We also
develop strong notion of graphic values where the existence of the
corresponding graphs is guaranteed once the constraints are satisfied by the
basic graph invariants. These constraints are necessary and sufficient for
graphs whose local neighborhoods are generated by a finite set of locally
connected graphs. The reconstruction of the graph from the basic graph
invariants is shown to be NP-complete in this restricted case. Finally we apply
these results to formulate the problem of Ramsey numbers as an integer
polyhedron problem of moderate and adjustable dimension.
