Graph states are special kinds of multipartite entangled states that correspond to mathematical graphs where the vertices take the role of quantum spin systems and the edges represent interactions. They not only provide an efficient model to study multiparticle entanglement, but also find wide applications in quantum error correction, multi-party quantum communication and most prominently, serve as the central resource in one-way quantum computation. Here we report the creation of two special instances of graph states, the six-photon Greenberger-Horne-Zeilinger states -- the largest photonic Schr\"{o}dinger cat, and the six-photon cluster states-- a state-of-the-art one-way quantum computer. Flexibly, slight modifications of our method allow creation of many other graph states. Thus we have demonstrated the ability of entangling six photons and engineering multiqubit graph states, and created a test-bed for investigations of one-way quantum computation and studies of multiparticle entanglement as well as foundational issues such as nonlocality and decoherence.