### Abstract

In Part I of the present paper the following problem was investigated. Let G be a finite simple graph, and S be a finite set of primes. We say that G is representable with S if it is possible to attach rational numbers to the vertices of G such that the vertices v_{1}, v_{2} are connected by an edge if and only if the difference of the attached values is an S-unit. In Part I we gave several results concerning the representability of graphs in the above sense. In the present paper we extend the results from Part I to the algebraic number field case and make some of them effective. Besides we prove some new theorems: we prove that G is infinitely representable with S if and only if it has a degenerate representation with S, and we also deal with the representability with S of the union of two graphs of which at least one is finitely representable with S.

Original language | English |
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Pages (from-to) | 423-447 |

Number of pages | 25 |

Journal | Acta Mathematica Hungarica |

Volume | 149 |

Issue number | 2 |

DOIs | |

Publication status | Published - Aug 1 2016 |

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### Keywords

- arithmetic graph
- cubical graph
- representability
- S-unit equation

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Acta Mathematica Hungarica*,

*149*(2), 423-447. https://doi.org/10.1007/s10474-016-0633-y