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

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