We show that a canonical labeling of a random $n$-vertex graph can be obtained by assigning to each vertex $x$ the triple $(w_1(x),w_2(x),w_3(x))$, where $w_k(x)$ is the number of walks of length $k$ starting from $x$. This takes time $O(n^2)$, where $n^2$ is the input size, by using just two matrix-vector multiplications. The linear-time canonization of a random graph is the classical result of Babai, Erd\H{o}s, and Selkow. For this purpose they use the well-known combinatorial color refinement procedure, and we make a comparative analysis of the two algorithmic approaches.

翻译：我们证明，随机 $n$ 顶点图的典范标号可以通过为每个顶点 $x$ 分配三元组 $(w_1(x), w_2(x), w_3(x))$ 获得，其中 $w_k(x)$ 表示从 $x$ 出发长度为 $k$ 的游走数量。该过程仅需两次矩阵-向量乘法，时间复杂度为 $O(n^2)$，其中 $n^2$ 为输入规模。随机图的线性时间典范化是 Babai、Erdős 和 Selkow 的经典成果。他们为此使用了著名的组合颜色细化过程，我们对这两种算法方法进行了比较分析。