A proof of Haemers' toughness conjecture

We prove that if $\mathrm{\Gamma }$ is a connected graph with minimum degree $\delta$ and Laplacian eigenvalues $0={\mu }_1<{\mu }_2⩽\cdots ⩽{\mu }_n$, then the toughness of $\mathrm{\Gamma }$ is bounded below by ${\mu }_2\mathrm{/}({\mu }_n-\delta )$.