图同构问题的组合不变量
Combinatorial invariants for graph isomorphism problem
摘要：
提出的方法在多项式时间内计算每个顶点的大量不变量，这些不变量不随图的同构而改变，并且应该完全确定图。例如，给定起始点的长度为k的闭合路径的数量，可以认为是邻接矩阵的k次幂的对角项。当k=2时，我们得到的是实点的度不变，越高，描述的局部拓扑就越深。现在，如果两个图是同构的，它们有相同的一组这样的不变量向量-我们可以对这些向量进行字典排序并比较它们。如果它们一致，排序的置换允许重构同构。我提出的论点是，这些不变量应该完全决定图，但不幸的是，我不能证明这一点在这一刻。这种方法可以带来希望，也许P=NP -而不是检查所有的实例，我们应该对这些大数字进行算术。
英文摘要：
Presented approach in polynomial time calculates large number of invariants for each vertex, which won't change with graph isomorphism and should fully determine the graph. For example numbers of closed paths of length k for given starting vertex, what can be though as the diagonal terms of k-th power of the adjacency matrix. For k=2 we would get degree of verities invariant, higher describes local topology deeper. Now if two graphs are isomorphic, they have the same set of such vectors of invariants - we can sort theses vectors lexicographically and compare them. If they agree, permutations from sorting allow to reconstruct the isomorphism. I'm presenting arguments that these invariants should fully determine the graph, but unfortunately I can't prove it in this moment. This approach can give hope, that maybe P=NP - instead of checking all instances, we should make arithmetics on these large numbers.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
