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Toughness in pseudo-random graphs
European Journal of Combinatorics ( IF 1.0 ) Pub Date : 2020-10-13 , DOI: 10.1016/j.ejc.2020.103255
Xiaofeng Gu

A d-regular graph on n vertices with the second largest absolute eigenvalue at most λ is called an (n,d,λ)-graph. The celebrated expander mixing lemma for (n,d,λ)-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(G) of a connected graph G is defined as t(G)=min{|S|c(GS)}, where c(GS) denotes the number of components of GS and the minimum is taken over all proper subsets SV(G) such that c(GS)>1. It has been shown that for any connected (n,d,λ)-graph G, t(G)>13(d2dλ+λ21) by Alon, and independently, t(G)>dλ2 by Brouwer. Brouwer also conjectured that a tight lower bound should be t(G)dλ1 for any (n,d,λ)-graph G. We show that t(G)>dλ2. 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,λ)-graphs, which implies an improved result for classical vertex connectivity.



中文翻译:

伪随机图中的韧性

一种 d-正则图 ñ 最多具有第二大绝对特征值的顶点 λ 被称为 ñdλ-图形。著名的扩展器混合引理ñdλ-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ŤG 连通图 G 被定义为 ŤG={|小号|CG-小号},在哪里 CG-小号 表示的组件数 G-小号 最小值将覆盖所有适当的子集 小号VG 这样 CG-小号>1个。已经证明对于任何连接ñdλ-图形 GŤG>1个3d2dλ+λ2-1个 由阿隆(Alon)独立完成, ŤG>dλ-2由布劳维尔。布劳维尔还猜想应该将紧下限ŤGdλ-1个 对于任何 ñdλ-图形 G。我们证明ŤG>dλ-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ñdλ-graphs,这意味着改进了经典顶点连接性。

更新日期:2020-10-15
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