Toughness in regular graphs from eigenvalues
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918246197755904 |
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| author | Liu, Ruifang Fan, Ao Shu, Jinlong |
| author_facet | Liu, Ruifang Fan, Ao Shu, Jinlong |
| contents | The {\it toughness} $τ(G)=\mathrm{min}\{\frac{|S|}{c(G-S)}: S~\mbox{is a vertex cut in}~G\}$ for $G\ncong K_n,$ which was initially proposed by Chvátal in 1973. A graph $G$ is called {\it $t$-tough} if $τ(G)\geq t.$ Let $λ_i(G)$ be the $i$-th largest eigenvalue of the adjacency matrix of a graph $G$. In 1996, Brouwer conjectured that $τ(G)\geq\frac{d}λ-1$ for a connected $d$-regular graph $G,$ where $λ=\mathrm{max}\{|λ_2|, |λ_n|\}.$ Gu [SIAM J. Discrete Math. 35 (2021) 948-952] completely confirmed this conjecture. From Brouwer and Gu's result $τ(G)\geq\frac{d}λ-1,$ we know that if $G$ is a connected $d$-regular graph and $λ\leq\frac{bd}{b+1}$, then $τ(G)\geq\frac{1}{b}$ for an integer $b\geq1.$ Inspired by the above result and utilizing typical spectral techniques and graph construction methods from Cioabă et al. [J. Combin. Theory Ser. B 99 (2009) 287-297], we prove that if $G$ is a connected $d$-regular graph and $λ_2(G)<ϕ(d,b)$, then $τ(G)\geq\frac{1}{b}$. Meanwhile, we construct graphs implying that the upper bound on $λ_2(G)$ is best possible. Our theorem strengthens the result of Chen et al. [Discrete Math. 348 (2025) 114404]. Finally, we also prove an upper bound of $λ_{b+1}(G)$ to guarantee a connected $d$-regular graph to be $\frac{1}{b}$-tough. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_07007 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Toughness in regular graphs from eigenvalues Liu, Ruifang Fan, Ao Shu, Jinlong Combinatorics 05C50, 05C35 The {\it toughness} $τ(G)=\mathrm{min}\{\frac{|S|}{c(G-S)}: S~\mbox{is a vertex cut in}~G\}$ for $G\ncong K_n,$ which was initially proposed by Chvátal in 1973. A graph $G$ is called {\it $t$-tough} if $τ(G)\geq t.$ Let $λ_i(G)$ be the $i$-th largest eigenvalue of the adjacency matrix of a graph $G$. In 1996, Brouwer conjectured that $τ(G)\geq\frac{d}λ-1$ for a connected $d$-regular graph $G,$ where $λ=\mathrm{max}\{|λ_2|, |λ_n|\}.$ Gu [SIAM J. Discrete Math. 35 (2021) 948-952] completely confirmed this conjecture. From Brouwer and Gu's result $τ(G)\geq\frac{d}λ-1,$ we know that if $G$ is a connected $d$-regular graph and $λ\leq\frac{bd}{b+1}$, then $τ(G)\geq\frac{1}{b}$ for an integer $b\geq1.$ Inspired by the above result and utilizing typical spectral techniques and graph construction methods from Cioabă et al. [J. Combin. Theory Ser. B 99 (2009) 287-297], we prove that if $G$ is a connected $d$-regular graph and $λ_2(G)<ϕ(d,b)$, then $τ(G)\geq\frac{1}{b}$. Meanwhile, we construct graphs implying that the upper bound on $λ_2(G)$ is best possible. Our theorem strengthens the result of Chen et al. [Discrete Math. 348 (2025) 114404]. Finally, we also prove an upper bound of $λ_{b+1}(G)$ to guarantee a connected $d$-regular graph to be $\frac{1}{b}$-tough. |
| title | Toughness in regular graphs from eigenvalues |
| topic | Combinatorics 05C50, 05C35 |
| url | https://arxiv.org/abs/2510.07007 |