on GY indicate their chemisorption characteristics. The weakest chemisorption case is Mn-GY whose adsorption energy is -0.907 eV, whereas the strongest chemisorption one is the Fe-GY whose adsorption energy is -2.914 eV. The stability of TM-chain-GY systems is derived from the existence of the carbon atoms with the -C$\equiv$C- bond as reported in our previous study[@19]. The valence electrons of the TM atom couple to not only the $p_z$ but also the $p_{x-y}$ of carbon atoms of the TM-chain-GY systems. The results in Tab.\[tab1\] indicate that as for the maximum (minimum) distance between the adatom of TM-chain and the nearest carbon atom ($d_{AC}$) is 2.29 $\AA$ (2.03 $\AA$) for V (Mn). All TM adatoms of TM-chain only slightly protrude out of the GY plane except Mn; the largest height (h) is 1.47 $\AA$ for V, whereas the minimum height is 0 $\AA$ for Mn. The result indicates that the size of Mn adatom matches the acetylene ring of GY very well.\

        dimer         V     Cr     Mn     Fe     Co
  ----------------- ------ ----- ------ ------ ------
     M($\mu_B$)      6.12   0.0   7.55   5.50   2.92
   $\Delta E$(meV)   -48    22    -122   -17     -8

  : The calculated results of TM-dimer-GY. M is the total magnetic moment per unit 