A Proof of Brouwer's Toughness Conjecture

The toughness $t(G)$ of a connected graph $G$ is defined as $t(G)=\min\{\frac{|S|}{c(G-S)}\}$, in which the minimum taken over all proper subsets $S\subset V(G)$ such that $c(G-S)>1$, where $c(G-S)$ denotes number components $G-S$. Let $\lambda$ denote second largest absolute eigenvalue adjacency matrix graph. For any $d$-regular $G$, it has been shown by Alon $t(G)>\frac{1}{3}(\frac{d^2}{d\lambda+\lambda^2}-1)$, through which, was able to show for every $t$ and $g$ there are $t$-tough graphs girth strictly greater than $g$, thus disproved strong sense conjecture Chv\'atal on pancyclicity. Brouwer independently discovered better bound $t(G)>\frac{d}{\lambda}-2$ while he also conjectured lower can be improved $t(G)\ge \frac{d}{\lambda} - 1$. We confirm this conjecture.