To understand the magnetic exchange coupling mechanism of TM-chain-GY, we show the partial density of states (PDOS) of TM and their nearest neighbor (NN) carbon atom in Fig.\[fig4\]. We only show the PDOS of one of the NN *sp*-hybridized carbon atoms of the TM adatom in the figure because the four NN carbon atoms of each TM adatom show similar characteristics. Here we take Fe-chains-GY as example to elucidate the relationship between the coupling and the magnetic properties of TM-chains-GY. The PDOS of Fe adatom and its NN carbon atoms for Fe-chain-GY are shown in Fig.\[fig4\](e). For the majority spin, the Fe *d* orbital with A1($d_{z^2}$), E1 ($d_{yz}$,$d_{xz}$) and E2 ($d_{xy}$,$d_{x^2+y^2}$) symmetry strongly couples with the in-plane $p_{x-y}$ and out-plane $p_z$ of its NN C atoms in the energy ranges of -6.4eV to -2.1 eV. For the minority spin, one can see that there is considerable overlap for the A1, E1 and E2 states of Fe atom and $p_{x-y}$ and $p_z$ states of NN C atoms within -2.8 eV to -1.2 eV energy window. Because of the interaction of $p_z-d$ and $p_{x-y}-d$ around the Fermi level, the majority-spin *p* band is shifted to higher energy, while the minority-spin *p* band 