It is confirmed in this work that the graph isomorphism can be tested in polynomial time, which resolves a longstanding problem in the theory of computation. The contributions are in three phases as follows. 1. A description graph $\tilde{A}$ to a given graph $A$ is introduced so that labels to vertices and edges of $\tilde{A}$ indicate the identical or different amounts of walks of any sort in any length between vertices in $A$. Three processes are then developed to obtain description graphs. They reveal relations among matrix power, spectral decomposition and adjoint matrices, which is of independent interest. 2. We show that the stabilization of description graphs can be implemented via matrix-power stabilization, a new approach to distinguish vertices and edges to graphs. The approach is proven to be equivalent in the partition of vertices to Weisfeiler-Lehman (WL for short) process. The specific Square-and-Substitution (SaS) process is more succinct than WL process. The vertex partitions to our stable graphs are proven to be \emph{strongly} equitable partitions, which is important in the proofs of our main conclusion. Some properties on stable graphs are also explored. 3. A class of graphs named binding graphs is proposed and proven to be graph-isomorphism complete. The vertex partition to the stable graph of a binding graph is the automorphism partition, which allows us to confirm graph-isomorphism problem is in complexity class $\mathtt{P}$. Since the binding graph to a graph is so simple in construction, our approach can be readily applied in practice.
翻译:这项工作证实, 图形是形态化的, 可以用多面性时间来测试, 从而解决计算理论中长期存在的问题。 贡献分为以下三个阶段。 1. 引入给给定图形$A$的描述图形$\ tilde{ A}$A$ 美元, 以便给顶端和边缘贴上标签 $\ tilde{ A} 美元 美元 表示任何长于任何长度的顶端。 然后开发三个进程, 以获取描述性图表。 它们显示矩阵动力、 光谱分解和 联合矩阵之间的关系, 这是独立感兴趣的 。 我们显示, 描述性图表的稳定性可以通过矩阵稳定化执行, 新的方法可以区分顶端和图形的边缘。 这个方法被证明相当于向 Weisfeiler- Lehman (WL 用于短期) 的顶端。 特定的平面和 平面图解( Sa) 进程比 WL 进程更简洁化 。 某些平面性分布 也是我们图表中稳定的稳定性 。 的稳定性 的稳定性 。 开始 。