To analyze spin polarization of the systems, we define the spin polarization $P(E_{f})$ at the Fermi level as: $$P(E_{f})=\frac{D(E_{f\uparrow})-D(E_{f\downarrow})}{D(E_{f\uparrow})+D(E_{f\downarrow})}$$ where $D(E_{f\uparrow})$ and $D(E_{f\downarrow})$ represent the density of states (DOS) of majority spin and minority spin at the Fermi level, respectively.\
To have a better understand of the magnetic coupling of the TM-GY systems, we use the map of spin-polarized charge density (SCD) for all investigated systems. The SCD are defined as: $$\rho(\vec r)=\rho\uparrow(\vec r)-\rho\downarrow(\vec r)$$ where $\rho\uparrow(\vec r)$ and $\rho\downarrow(\vec r)$ are the spin up and the spin down charge density of TM-GY system, respectively.\

        h($\AA$)   $d_{AC}$($\AA$)   M($\mu_B$)   $\Delta E(meV)$   $E_{a}(eV)$   $T_e$    P    MS
  ---- ---------- ----------------- ------------ ----------------- ------------- ------- ----- -----
   V      1.47          2.29            7.32           -271           -2.787      1.16    63%   FM
   Cr     1.24          2.26             0              42            -1.190      1.01    0 %   AFM
   Mn      0            2.03            7.63           -193           -0.907      1.28    12%   FM
   Fe     0.74          2.14            5.67            -11           -2.914      0.94    14%   FM
   Co     0.49          2.12            3.5             -16           -2.108      0.76    24%   FM

Results and discussions
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Adsorption Geometry and Stability of TM chain on the surface of GY
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Before investigating the magnetic properties of TM-chains-GY systems, we study the stability of TM chains adsorbed GY systems. First of all , we considered 