On the Simple $\Z_2$-homotopy Types of Graph Complexes and Their Simple $\Z_2$-universality We prove that the neighborhood complex $\N(G)$, the box complex $\B(G)$, the homomorphism complex $\Hom(K_2,G)$and the Lov\'{a}sz complex $\L(G)$ have the same simple $\Z_2$-homotopy type in the sense of Whitehead. We show that these graph complexes are simple $\Z_2$-universal. Keywords:graph complexes, simple $\Z_2$-homotopy, universalityCategories:57Q10, 05C10, 55P10