A dynamical system of points moving along the edges of a graph could be considered as a geometrical discrete dynamical system or as a discrete version of a quantum graph with localized wave packets. We study the set of such systems over metric graphs that can be constructed from a given set of commensurable edges with fixed lengths. It is shown that there always exists a system consisting of a bead graph with vertex degrees not greater than three that demonstrates the longest stabilization time in such a set. The results are extended to graphs with incommensurable edges using the notion of $\varepsilon$-nets and, also, it is shown that dynamical systems of points on linear graphs have the slowest growth of the number of dynamic points

翻译：沿着图形边缘移动的点的动态系统可被视为几何离散动态系统,或带有局部波包的量子图离异版本。我们研究了从一组固定长度的可相互对应的边缘组成的一套系统,以图形图为基础,用一套固定长度的可比较边缘构建的一套系统。可以表明,始终存在由一个顶部不大于3度的珠形图组成的系统,显示在一组中稳定时间最长。结果被扩展至使用 $\ varepsilon$-net 概念具有不可比较边缘的图表,并且还显示,线形图上的点动态系统增长最慢,动态点数增长最慢。