The Laplacian S_{n,n} conjecture is true

Abstract:
The “Sn,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 ≥ 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.
Authors:

  • Nathaniel Johnston

Download:

  • Preprint from arXiv:2609.26895 [math.CO]

Cite as:

  • N. Johnston. The Laplacian Sn,n conjecture is true. E-print: arXiv:2609.26895 [math.CO], 2026.