For a given graph G with n vertices, the walk matrix W of G is defined as $W=\left[\begin{array}{ccccc}\hfill e\hfill & \hfill Ae\hfill & \hfill A^{2}e\hfill & \hfill \cdots \hfill & \hfill A^{n-1}e\hfill \end{array}\right]$, where A is the adjacency matrix of the graph G and e is the vector of all ones. In this paper, we prove that for any positive integer k, at most $\lfloor \frac{n}{2}\rfloor$ invariant factors of W are congruent to $2^k$ modulo $2^{k+1}$.

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关于图游走矩阵不变因子的注解

对于给定的图表ģ与Ñ顶点，步行矩阵W¯¯的ģ被定义为$宽=\left[\begin{array}{ccccc}\hfill \mathrm{电子}\hfill & \hfill \mathrm{一种}\mathrm{电子}\hfill & \hfill {\mathrm{一种}}^2\mathrm{电子}\hfill & \hfill \cdots \hfill & \hfill {\mathrm{一种}}^{n-1}\mathrm{电子}\hfill \end{array}\right]$，其中阿是图的邻接矩阵ģ和Ë是所有那些的载体。在本文中，我们证明对于任意正整数k，至多$\lfloor \frac{n}{2}\rfloor$W 的不变因子与$2^{克}$ 模数 $2^{克+1}$.