图不变量环-图值
The Ring of Graph Invariants - Graphic Values
摘要：
图不变量环由基本图不变量构成，基本图不变量计算与给定图同构的子图在其它图中的个数。这些子图计数不变量不是代数独立的。在我们看来，无标号图的图论中最重要的问题是确定任意图不变量集的图值的问题。这对应于解释当顶点数无界时图不变量的合合。我们介绍两种方法来探索这个复杂的结构。具有少量顶点的图的集合对较大的集合施加约束。我们描述家庭的不变量图的不等式。这些不等式允许循环所有的图不变量的值，这些不变量从小集合的角度来看看起来像图形。我们还开发了强有力的概念，图形值的存在相应的图形是保证一旦满足基本图形不变量的约束。对于局部邻域由局部连通图的有限集生成的图，这些约束是必要且充分的。图的重建从基本的图不变量示出在这种限制的情况下是NP完全的。最后，我们应用这些结果制定的问题拉姆齐数的整数多面体问题的适度和可调的维数。
英文摘要：
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.
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