cell. $\Delta E$ is the energy difference defined as $\Delta E$=$E_{FM}$-$E_{AFM}$, where $E_{FM}$ and $E_{AFM}$ are the total energies for ferromagnetic and antiferromagnetic states in the same unit cell, respectively.[]{data-label="tabdimer"}

Magnetic Order and Electronic Structures of TM-chain-GY
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On the basis of the most stable structures of TM-dimer-GY and TM-chain-GY, we systematically investigate their magnetic exchange coupling and electronic properties. Distinct spin polarization is found for the GY with adsorption of V, Cr, Mn, Fe, and Co dimer and chain. We consider only a collinear ferromagnetic (FM) and antiferromagnetic (AFM) coupling for each TM-dimer-GY and TM-chain-GY configuration at present. The exchange energy ($\Delta E$=$E_{FM}$-$E_{AFM}$) between the FM and AFM states per unit cell is calculated and shown in Tab.\[tabdimer\] and Tab.\[tab1\] for TM-dimer-GY and TM-chain-GY, respectively. Positive (negative) exchange energy indicates that the ground state of the system is AFM (FM). For the case of TM-dimer-GY, the two TM adatoms for V, Mn, Fe, and Co show ferromagnetic coupling characteristics, whereas Cr dimer shows antiferromagnetic coupling characteristics with zero magnetic moment. Moreover, for V and Mn dimer, the exchange energy is larger enough to stabilize their ferromagnetic nature at high temperature. Such magnetic coupling for TM-dimer-GY is derived form the 