Toughness in pseudo-random graphs

Abstract

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(G−S)}, where c(G−S) denotes the number of components of G−S and the minimum is taken over all proper subsets S⊂V(G) such that c(G−S)>1. It has been shown that for any connected (n,d,λ)-graph G, t(G)>13(d2dλ+λ2−1) by Alon, and independently, t(G)>dλ−2 by Brouwer. Brouwer also conjectured that a tight lower bound should be t(G)≥dλ−1 for any (n,d,λ)-graph G. We show that t(G)>dλ−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,λ)-graphs, which implies an improved result for classical vertex connectivity.