Corrigendum to the equivalent statement of the Laplacian Spread Conjecture

TL;DR

This paper corrects previous conclusions about the Laplacian Spread Conjecture, providing precise conditions for eigenvalue inequalities and bounds for balanced digraphs, refining earlier inaccuracies.

Contribution

It clarifies the necessary and sufficient conditions for the Laplacian eigenvalue inequality and corrects bounds on the Laplacian spread of balanced digraphs.

Findings

Corrected the condition for the Laplacian eigenvalue sum inequality.

Refined the upper bound for the Laplacian spread of balanced digraphs.

Disproved the previous inequality involving the squared norm condition.

Abstract

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