A proof of Brouwer's toughness conjecture

Abstract

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, Alon 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 Chv\'atal 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.