A linear algebraic proof of the Laplacian spread conjecture

Summary · qwen2.5:32b

The article presents a novel and more concise linear algebraic proof for the Laplacian Spread Conjecture, stating that for any graph $G$, the sum of its second smallest Laplacian eigenvalue and that of its complement $\overline{G}$ is at least 1. This proof offers a significant advancement in graph theory by confirming the conjecture with a different mathematical approach.

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In this new paper, we explore the Laplacian Spread Conjecture in graph theory using linear algebra. It's a fascinating approach to prove such a conjecture in the field of rendering and graphics.

Excerpt

arXiv:2607.03711v1 Announce Type: new 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.