Long, stable, and free-standing linear atomic carbon wires (carbon chains) have been carved out from graphene recently [Meyer et al. Nature (London) 2008, 454, 319; Jin et al. Phys. Rev. Lett. 2009, 102, 205501]. They can be considered as extremely narrow graphene nanoribbons or extremely thin carbon nanotubes. It might even be possible to make use of high-strength and identical (without chirality) carbon wires as a transport channel or on-chip interconnects for field-effect transistors. Here we investigate electron transport properties of linear atomic carbon wire-graphene junctions by combining nonequilibrium Green's function with density functional theory. For short wires, linear ballistic transport is observed in wires consisting of odd numbers of carbon atoms but not in those consisting of even numbers of carbon atoms. For wires longer than 2.1 nm as fabricated above, however, the ballistic conductance of carbon wire-graphene junctions is independent of the structural distortion, structural imperfections, and hydrogen impurity adsorbed on the linear carbon wires, except for oxygen impurity adsorption under a low bias. As such, the epoxy groups might be the origin of experimentally observed low conductance in the carbon chain. Moreover, double-atomic carbon chains exhibit a negative differential resistance effect.