Towards Dynamic-Point Systems on Metric Graphs with Longest Stabilization Time

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