The Ring of Graph Invariants - Graphic Values

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. Sets of basic graph invariants form ensembles where each graph in the set induces the corresponding invariant calculating the number of subgraphs isomorphic to this graph in other graphs. It is well known that all other graph invariants such as sorted eigenvalues and canonical permutations are linear combinations of the basic graph invariants. 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. Ensembles with a small number of vertices impose constraints on larger ensembles. 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 ensembles point of view. The inequalities give rise to a weak notion of graphic values where the existence of the corresponding graph is not guaranteed. 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. In fact all extremal graph problems are formulated as integer programming problems.