Ballistic transport through carbon atomic chain (CAC) attached to relaxed armchair graphene (ACG), zigzag graphene (ZZG), and unrelaxed armchair graphene (UACG) electrodes have been investigated, as a function of energy measured from Fermi level with the use of PWSCF code of Quantum Espresso package. Our computed results show that the equilibrium conductance of CAC attached to ZZG-electrodes tends to zero over a wide energy range across Fermi level. Similar behaviour of conductance is not seen when ZZG electrodes are replaced by ACG electrodes. CAC connected to ACG electrode is found better conducting as compared to CAC connected to ZZG electrode. The computed conductance, for a fixed energy value, strongly depends on whether CAC is attached to ZZG, ACG or UACG electrodes, suggesting that transport through electrode–CAC–electrode system depends on electrode edge structure. On comparing conductance of CAC connecting to graphene electrodes with that of CAC attached to metallic (Al) electrodes, we find that replacement of metallic electrode by graphene makes conductance smoother over wider energy range.