SIAM Journal on Discrete Mathematics, Volume 35, Issue 2, Page 948-952, January 2021.
The toughness $t(G)$ of a connected graph $G$ is defined as $t(G)=\min\{\frac{|S|}{c(G-S)}\}$, in which the minimum is taken over all proper subsets $S\subset V(G)$ such that $c(G-S)>1$, where $c(G-S)$ denotes the number of components of $G-S$. Let $\lambda$ denote the second largest absolute eigenvalue of the adjacency matrix of a graph. For any connected $d$-regular graph $G$, it has been shown by Alon that $t(G)>\frac{1}{3}(\frac{d^2}{d\lambda+\lambda^2}-1)$, through which he was able to show that for every $t$ and $g$ there are $t$-tough graphs of girth strictly greater than $g$ and thus disproved in a strong sense a conjecture of Chvátal on pancyclicity. Brouwer independently discovered a better bound $t(G)>\frac{d}{\lambda}-2$ for any connected $d$-regular graph $G$, while he also conjectured that the lower bound can be improved to $t(G)\ge \frac{d}{\lambda} - 1$. We confirm this conjecture.

中文翻译：

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Brouwer 韧性猜想的证明

SIAM 离散数学杂志，第 35 卷，第 2 期，第 948-952 页，2021 年 1 月。
连通图$G$的韧性$t(G)$定义为$t(G)=\min\{\frac{|S|}{c(GS)}\}$，其中最小值为接管所有真子集$S\subset V(G)$，使得$c(GS)>1$，其中$c(GS)$表示$GS$的分量数。让 $\lambda$ 表示图的邻接矩阵的第二大绝对特征值。对于任何连通的 $d$-正则图 $G$，Alon 已经证明 $t(G)>\frac{1}{3}(\frac{d^2}{d\lambda+\lambda^2 }-1)$，通过它他能够证明对于每个 $t$ 和 $g$ 都有 $t$-tough 的周长图严格大于 $g$，因此在强烈意义上反驳了 Chvátal 的猜想关于泛周期性。Brouwer 独立地发现了一个更好的边界 $t(G)>\frac{d}{\lambda}-2$ 对于任何连接的 $d$-正则图 $G$，同时他还推测下界可以改进为$t(G)\ge \frac{d}{\lambda} - 1$。我们证实了这个猜想。