### Abstract

The discrete Toda molecule equation can be used to compute eigenvalues of tridiagonal matrices over conventional linear algebra, and is the recursion formula of the well-known quotient difference algorithm for tridiagonal eigenvalues. An ultradiscretization of the discrete Toda equation leads to the ultradiscrete Toda (udToda) equation, which describes motions of balls in the box and ball system. In this paper, we associate the udToda equation with eigenvalues of tridiagonal matrices over min-plus algebra, which is a semiring with two operation types: ⊕:= min and ⊗ := +. We also clarify an interpretation of the udToda variables in weighted and directed graphs consisting of vertices and edges.

Original language | English |
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Article number | 444001 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 51 |

Issue number | 44 |

DOIs | |

Publication status | Published - 2018 Oct 8 |

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### Keywords

- eigenvalue
- min-plus algebra
- ultradiscrete Toda equation
- weighted directed graph

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Statistics and Probability
- Modelling and Simulation
- Mathematical Physics
- Physics and Astronomy(all)

### Cite this

*Journal of Physics A: Mathematical and Theoretical*,

*51*(44), [444001]. https://doi.org/10.1088/1751-8121/aae325