Let $G$ be a graph on $n$ vertices with adjacency matrix $A$, and let $\mathbf{1}$ be the all-ones vector. We call $G$ controllable if the set of vectors $\mathbf{1}, A\mathbf{1}, \dots, A^{n-1}\mathbf{1}$ spans the whole space $\mathbb{R}^n$. We characterize the isomorphism problem of controllable graphs in terms of other combinatorial, geometric and logical problems. We also describe a polynomial time algorithm for graph isomorphism that works for almost all graphs.

翻译：设 $G$ 为具有邻接矩阵 $A$ 的 $n$ 个顶点图，$\mathbf{1}$ 为全一向量。若向量集 $\mathbf{1}, A\mathbf{1}, \dots, A^{n-1}\mathbf{1}$ 张成整个空间 $\mathbb{R}^n$，则称 $G$ 为可控图。我们从组合、几何与逻辑问题的角度刻画了可控图的同构问题。此外，我们还描述了一种对几乎所有图均有效的多项式时间图同构算法。