Corrigendum to the equivalent statement of the Laplacian Spread Conjecture

For a graph $G,$ let $\alpha (G)$ denote its second smallest Laplacian eigenvalue. The Laplacian Spread Conjecture states that $\alpha (G)+\alpha (\bar{G})\ge 1,$ where $\bar{G}$ is the complement of $G\mathrm{.}$ In this paper, we have corrected two conclusions: First, the necessary and sufficient condition for $\alpha (G)+\alpha (\bar{G})\ge 1$ is $\parallel {▽}_x-{▽}_y{\parallel }^2\ge 1$ rather than $\parallel {▽}_x-{▽}_y{\parallel }^2\ge 2$ which has been proved in \cite{BS} as demonstrated in our study. Second, we show that the Laplacian spread of balanced digraph $\mathrm{\Gamma }$ satisfies $LS(\mathrm{\Gamma })\le n-\frac{1}{2}$ but not $LS(\mathrm{\Gamma })\le n-1$ in \cite{BCEHK}, since inequality $\parallel {▽}_x-{▽}_y{\parallel }^2\ge 2$ does not hold.