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The periodic discrete Toda equation defined over finite fields has beenstudied. We obtained the finite graph structures constructed by the network ofstates where edges denote possible time evolutions. We simplify the graphs byintroducing a equivalence class of cyclic permutations to the initial values.We proved that the graphs are bi-directional and that they are composed ofseveral arrays of complete graphs connected at one of their vertices.

Graphs emerging from the solutions to the periodic discrete Toda equation over finite fields