The toughness $t(G)$ of a graph G is a measure of its connectivity that is closely related to Hamiltonicity. Xiaofeng Gu, confirming a longstanding conjecture of Brouwer, recently proved the lower bound $t(G)\ge \ell /\lambda -1$ on the toughness of any connected ℓ-regular graph, where λ is the largest nontrivial absolute eigenvalue of the adjacency matrix. Brouwer had also observed that many families of graphs (in particular, those achieving equality in the Hoffman ratio bound for the independence number) have toughness exactly $\ell /\lambda$. Cioabă and Wong confirmed Brouwer's observation for several families of graphs, including Kneser graphs $K(n,2)$ and their complements, with the exception of the Petersen graph $K(5,2)$. In this paper, we extend these results and determine the toughness of Kneser graphs $K(n,k)$ when $k\in \{3,4\}$ and $n\ge 2k+1$ as well as for $k\ge 5$ and sufficiently large n (in terms of k). In all these cases, the toughness is attained by the complement of a maximum independent set and we conjecture that this is the case for any $k\ge 5$ and $n\ge 2k+1$.

该韧性 $吨(G)$图G的连通性度量与哈密顿性密切相关。顾晓峰，证实了 Brouwer 长期以来的猜想，最近证明了下界$吨(G)\ge \ell /\lambda -1$关于任何连通ℓ -正则图的韧性，其中λ是邻接矩阵的最大非平凡绝对特征值。Brouwer 还观察到，许多图族（特别是那些在 Hoffman 比上达到独立数界限的图）完全具有韧性$\ell /\lambda$. Cioabă 和 Wong 证实了 Brouwer 对几个图族的观察，包括 Kneser 图$钾(n,2)$ 及其补充，彼得森图除外 $钾(5,2)$. 在本文中，我们扩展了这些结果并确定了 Kneser 图的韧性$钾(n,克)$ 什么时候 $克\in \{3,4\}$ 和 $n\ge 2克+1$ 以及为了 $克\ge 5$和足够大的n（就k 而言）。在所有这些情况下，韧性是通过最大独立集的补集获得的，我们推测这对于任何$克\ge 5$ 和 $n\ge 2克+1$.