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arXiv · 2609.26895

The Laplacian $S_{n,n}$ conjecture is true

Nathaniel Johnston
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Abstract

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

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Nathaniel Johnston. 2026-09-22. The Laplacian $S_{n,n}$ conjecture is true. https://arxiv.org/abs/2609.26895

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