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

TL;DR

This paper investigates the structure of graphs generated by solutions to the periodic discrete Toda equation over finite fields, revealing their bi-directional nature and composition of interconnected complete graph arrays.

Contribution

It introduces a method to simplify these graphs using equivalence classes of cyclic permutations and characterizes their overall structure.

Findings

Graphs are bi-directional.

Graphs consist of arrays of complete graphs connected at vertices.

Simplification via cyclic permutation equivalence classes is effective.

Abstract

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