Corrigendum to the equivalent statement of the Laplacian Spread Conjecture
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913582330937344 |
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| author | Li, Borchen |
| author_facet | Li, Borchen |
| contents | For a graph $G,$ let $α(G)$ denote its second smallest Laplacian eigenvalue. The Laplacian Spread Conjecture states that $α(G)+α(\overline{G}) \geq 1,$ where $\overline{G}$ is the complement of $G.$ In this paper, we have corrected two conclusions: First, the necessary and sufficient condition for $α(G) + α(\overline{G}) \geq 1$ 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 $Γ$ satisfies $LS(Γ) \leq n - \frac{1}{2}$ but not $LS(Γ) \leq n - 1$ in \cite{BCEHK}, since inequality $\parallel \bigtriangledown_{x} - \bigtriangledown_{y} \parallel^{2} \geq 2$ does not hold. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_02440 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Corrigendum to the equivalent statement of the Laplacian Spread Conjecture Li, Borchen Combinatorics For a graph $G,$ let $α(G)$ denote its second smallest Laplacian eigenvalue. The Laplacian Spread Conjecture states that $α(G)+α(\overline{G}) \geq 1,$ where $\overline{G}$ is the complement of $G.$ In this paper, we have corrected two conclusions: First, the necessary and sufficient condition for $α(G) + α(\overline{G}) \geq 1$ 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 $Γ$ satisfies $LS(Γ) \leq n - \frac{1}{2}$ but not $LS(Γ) \leq n - 1$ in \cite{BCEHK}, since inequality $\parallel \bigtriangledown_{x} - \bigtriangledown_{y} \parallel^{2} \geq 2$ does not hold. |
| title | Corrigendum to the equivalent statement of the Laplacian Spread Conjecture |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2411.02440 |