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PreprintarXiv licencearXiv

The Laplacian Sn,n conjecture is true

  • Nathaniel Johnston
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Abstract

The "Sn,n conjecture" asserts that there does not exist a simple graph on n vertices with Laplacian spectrum {0,1,2,…,n−1} for any integer n ≥ 2. This conjecture has already been proved when 2 ≤ n ≤ 15 and when n ≥ 6,649,688,933. We prove all remaining cases and thus establish that the conjecture is true.

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Cite

BibTeXRIS
BibTeX
@misc{johnston2026laplacian,
  title = {{The Laplacian $S_n,n$ conjecture is true}},
  author = {Nathaniel Johnston},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.26895},
  eprint = {2609.26895},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.26895},
}
RIS
TY  - UNPB
TI  - The Laplacian S_n,n conjecture is true
AU  - Nathaniel Johnston
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.26895
UR  - https://arxiv.org/abs/2609.26895
AB  - The "S_n,n conjecture" asserts that there does not exist a simple graph on n vertices with Laplacian spectrum {0,1,2,…,n−1} for any integer n ≥ 2. This conjecture has already been proved when 2 ≤ n ≤ 15 and when n ≥ 6,649,688,933. We prove all remaining cases and thus establish that the conjecture is true.
ER  -