The spin magnetic moment under easy axis of magnetization, orbital magnetic moment under both easy and hard axis of magnetization for TM chain are calculated and the results are listed in Tab.\[tab3\]. For easy axis of magnetization, the sequence of orbital magnetic moment of TM adatom are Co$>$Fe$>$V$>$Mn$>$Cr. Especially, Fe and Co shown considerable orbital magnetic moment with 0.112 $\mu_B$ and the 0.230 $\mu_B$. Moreover, the orbital magnetic moment of Co-chain-GY is similar to that of Co chain on the Pt(111) surface[@52] and Co dimer adsorbed on graphene[@53; @54]. To characterize the MAE of TM-chain-GY systems, we perform calculations of the total energy while rotating the magnetization orientation (the magnetization orientation are defined by azimuth angle $\theta$ and polar angle $\phi$ as shown in Fig.\[fig1\], $\theta$ and $\phi$ use the z axis and x axis, respectively, as reference Cartesian axes): (i) from *x* axis (90,90) to *z* axis(0,90) within xz plane; (ii) from *y* axis (90,0) to *z* axis(90,90) within yz plane; (iii) from *x* axis (90,90) to *z* axis(90,0) within xy plane by 10 degree intervals with the definition of MAE per unit cell as follows: $$MAE_{xz}=E(90,90)-E(\theta,90),  \   \   \theta\in(90,0)$$ $$MAE_{yz}=E(90,0)-E(\theta,0), \   \   \theta\in(90,0)$$ $$MAE_{xy}=E(90,90)-E(90,\phi), \    \   \phi\in(90,0)$$ where 