Consider an unweighted, directed graph $G$ with the diameter $D$. In this paper, we introduce the framework for counting cycles and walks of given length in matrix multiplication time $\widetilde{O}(n^\omega)$. The framework is based on the fast decomposition into Frobenius normal form and the Hankel matrix-vector multiplication. It allows us to solve the All-Nodes Shortest Cycles, All-Pairs All Walks problems efficiently and also give some improvement upon distance queries in unweighted graphs.

翻译：考虑一个无权重有向图 $G$，其直径为 $D$。本文提出了一种在矩阵乘法时间 $\widetilde{O}(n^\omega)$ 内计算给定长度环与游走的框架。该框架基于Frobenius标准型的快速分解与Hankel矩阵-向量乘法。它使我们能够高效地解决全节点最短环、全对全游走问题，并在无权图的距离查询方面取得一定改进。