Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Author: Johnston, Nathaniel
Institution: Mount Allison University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214230211