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An atomic carbon chain joining two graphene flakes was recently realized in aground-breaking experiment by Jin {\it et al.}, Phys. Rev. Lett. {\bf 102},205501 (2009). We present {\it ab initio} results for the electron transportproperties of such chains and demonstrate complete spin-polarization of thetransmission in large energy ranges. The effect is due to the spin-polarizedzig-zag edge terminating each graphene flake causing a spin-splitting of thegraphene $\pi_z$ bands, and the chain states. Transmission occurs when thegraphene $\pi$-states resonate with similar states in the strongly hybridizededges and chain. This effect should in general hold for any $\pi$-conjugatedmolecules bridging the zig-zag edges of graphene electrodes. The polarizationof the transmission can be controlled by chemically or mechanically modifyingthe molecule, or by applying an electrical gate.

Atomic carbon chains as spin-transmitters: an \textit{Ab initio} transport study