A linear algebraic proof of the Laplacian spread conjecture

Bocheng Li; Qingzhong Ji

A linear algebraic proof of the Laplacian spread conjecture

Combinatorics 2026-07-04 v1

Authors: Bocheng Li , Qingzhong Ji

Abstract

For a graph $G,$ let $\alpha(G)$ denote its second smallest Laplacian eigenvalue. The Laplacian Spread Conjecture is that $\alpha(G)+\alpha(\overline{G}) \geq 1,$ where $\overline{G}$ is the complement of $G.$ In this article, we provide a new proof of the Laplacian spread conjecture by means of linear algebra, which is more concise.

Keywords

Laplacian spread Laplacian eigenvalue graph complement linear algebra conjecture proof

Cite

@article{arxiv.2607.03711,
  title  = {A linear algebraic proof of the Laplacian spread conjecture},
  author = {Bocheng Li and Qingzhong Ji},
  journal= {arXiv preprint arXiv:2607.03711},
  year   = {2026}
}