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Equal Lagrange Numbers with Non-Isomorphic Band Graphs: A Counterexample to Schiffler's Problem 6.3

  • Qiyue Tang
  • Yizhi Zhang
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Abstract

We consider the following question: if two lattice paths in the same set π’Ÿ(a,b) have the same Lagrange number, must their band graphs be isomorphic? We exhibit two explicit lattice paths in π’Ÿ(17,9) with the same Lagrange number but non-isomorphic associated band graphs, thereby giving a counterexample to this question.

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Cite

BibTeXRIS
BibTeX
@misc{tang2026equal,
  title = {{Equal Lagrange Numbers with Non-Isomorphic Band Graphs: A Counterexample to Schiffler's Problem 6.3}},
  author = {Qiyue Tang and Yizhi Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.18889},
  eprint = {2608.18889},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.18889},
}
RIS
TY  - UNPB
TI  - Equal Lagrange Numbers with Non-Isomorphic Band Graphs: A Counterexample to Schiffler's Problem 6.3
AU  - Qiyue Tang
AU  - Yizhi Zhang
PY  - 2026
DA  - 2026/10/02
PB  - arXiv
DO  - 10.48550/arxiv.2608.18889
UR  - https://arxiv.org/abs/2608.18889
AB  - We consider the following question: if two lattice paths in the same set π’Ÿ(a,b) have the same Lagrange number, must their band graphs be isomorphic? We exhibit two explicit lattice paths in π’Ÿ(17,9) with the same Lagrange number but non-isomorphic associated band graphs, thereby giving a counterexample to this question.
ER  -