A $d$-regular graph on $n$ vertices with the second largest absolute eigenvalue at most $\lambda$ is called an $(n,d,\lambda )$-graph. The celebrated expander mixing lemma for $(n,d,\lambda )$-graphs builds a connection between graph spectrum and edge distribution. In this paper, we present some applications of the expander mixing lemma. In particular, we make progress toward the toughness conjecture of Brouwer. The toughness $t\left(G\right)$ of a connected graph $G$ is defined as $t\left(G\right)=min\left\{\frac{\left|S\right|}{c(G-S)}\right\}$, where $c(G-S)$ denotes the number of components of $G-S$ and the minimum is taken over all proper subsets $S\subset V\left(G\right)$ such that $c(G-S)>1$. It has been shown that for any connected $(n,d,\lambda )$-graph $G$, $t\left(G\right)>\frac{1}{3}(\frac{d^2}{d\lambda +{\lambda }^2}-1)$ by Alon, and independently, $t\left(G\right)>\frac{d}{\lambda }-2$ by Brouwer. Brouwer also conjectured that a tight lower bound should be $t\left(G\right)\ge \frac{d}{\lambda }-1$ for any $(n,d,\lambda )$-graph $G$. We show that $t\left(G\right)>\frac{d}{\lambda }-\sqrt{2}$. As the generalized vertex connectivity is closely related to graph toughness, we also present a lower bound on the generalized vertex connectivity of $(n,d,\lambda )$-graphs, which implies an improved result for classical vertex connectivity.

一种 $d$-正则图 $ñ$ 最多具有第二大绝对特征值的顶点 $\lambda$ 被称为 $（ñ，d，\lambda ）$-图形。著名的扩展器混合引理$（ñ，d，\lambda ）$-graphs在图谱和边缘分布之间建立联系。在本文中，我们介绍了膨胀器混合引理的一些应用。特别是，我们在Brouwer的韧性猜想方面取得了进展。韧性$Ť（G）$ 连通图 $G$ 被定义为 $Ť（G）=分\left\{\frac{\left|\mathrm{小号}\right|}{C（G-\mathrm{小号}）}\right\}$，在哪里 $C（G-\mathrm{小号}）$ 表示的组件数 $G-\mathrm{小号}$ 最小值将覆盖所有适当的子集 $\mathrm{小号}\subset V（G）$ 这样 $C（G-\mathrm{小号}）>\mathrm{1个}$。已经证明对于任何连接$（ñ，d，\lambda ）$-图形 $G$， $Ť（G）>\frac{\mathrm{1个}}{3}（\frac{d^2}{d\lambda +{\lambda }^2}-\mathrm{1个}）$ 由阿隆（Alon）独立完成， $Ť（G）>\frac{d}{\lambda }-2$由布劳维尔。布劳维尔还猜想应该将紧下限$Ť（G）\ge \frac{d}{\lambda }-\mathrm{1个}$ 对于任何 $（ñ，d，\lambda ）$-图形 $G$。我们证明$Ť（G）>\frac{d}{\lambda }-\sqrt{2}$。由于广义顶点连通性与图韧性密切相关，因此我们还给出了$（ñ，d，\lambda ）$-graphs，这意味着改进了经典顶点连接性。