shown in Fig.\[fig4\](a) indicates that the highest occupied (HOMO) and the lowest unoccupied (LUMO) *d* orbital are both in spin up channel under FM state; whereas under AFM state, the HOMO is spin-up E1 and E2 states, and the LUMO is spin-down E1 and E2 states. Therefore, the favorable magnetic state of the V-chain-GY is FM due to virtual hopping between the frontier orbital and Hund’s rules. The systems Mn-chain-GY, Fe-chain-GY, and Co-chain-GY have the similar virtual hopping characteristics; all of them show FM ground state. On the contrary, the virtual hopping for Cr-chain-GY, as shown in Fig.\[fig5\](b), will lead the system to AFM order. The results of virtual hopping of the energy levels between neighboring TM atoms are agree well with the above results.\
As mentioned above, TM-chain-GY systmes show strong coupling between TM-$d$ states and both out-plane $p_z$ and in-plane $p_{x-y}$ states of carbon atoms with $sp$ hybridization. Such coupling will induce electron exchange between TM and GY. To evaluate the charge transfer between the TM adatom and the GY sheet, the atomic basin charge based on the atoms in molecular (AIM) theory[@45] is adopted. By comparing the valence electrons in TM atomic basin of TM-chain-GY systems and 