A linear algebraic proof of the Laplacian spread conjecture

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