only the GGA results are presented below. The calculated electronic band structures along the high symmetry directions in the Brillouin zone are shown in Fig. 2. As discussed above, for InNCo$_{3}$ the ferromagnetic state is energetically preferable to the paramagnetic state, and hence it is clearly seen that the spin-splitting occurs in the bands around the E$_{F}$ . For InNCo$_{3}$ the profile of majority spin bands looks roughly similar to the one of minority spin bands, however only one band in the majority spin bands crosses the Fermi level and the corresponding band with same dispersion in the minority spin bands is unoccupied. In addi- tion, two bands in the minority spin bands across the Fermi level of InNCo$_{3}$ (see Fig. 2(a)). For InNNi$_{3}$ the ferromagnetic state is not energetically preferable to the paramagnetic state, consequently, the majority and minority spin bands are degenerated, that is to say, no spin- splitting occurs in the band structure (see Fig. 2(b)). It is interesting to note that the whole feature of majority spin bands of InNNi$_{3}$ is very similar to the one of InNCo$_{3}$. In order to reveal the detailed character of band structure, the total density of states (DOS) and the angular-momentum-projected DOS of each atom in InNCo$_{3}$ and InNNi$_{3}$ are presented in Fig. 3. For both InNCo$_{3}$ and InNNi$_{3}$, the four bands from -9 eV to -5 eV come mainly from the N 2 p states and In 5 s states, and the five bands roughly from -5 eV to -2.5 eV have significant contribution from 3 d - t$_{2}$$_{g}$ states of transition metal atoms (Co/Ni) and the 5 p state of In atom. For the bands from -2.5 eV to 0 eV (i.e., the Fermi level), they are dominated by the 3 d states of transition metal atoms (Co/Ni) and have small contribution from the 2 p states of N atom. Because the number of 3 d electrons in Co is one less than that of Ni, two minority spin bands composed of the 3 d - t$_{2}$$_{g}$ states around the Fermi level are unoccupied in InNCo$_{3}$, while the counterpart in the InNNi$_{3}$ are occupied. This also results in different behavior of spin-splitting in InNCo$_{3}$ and InNNi$_{3}$. Due to the significant spin-plitting around the Fermi level in InNCo$_{3}$, the hybridization between the Co 3 d and N 2 p states in InNCo$_{3}$ are slightly weaker than the one between Ni 3 d and N 2 p states in InNNi$_{3}$(see Fig. 3). The contributions of each kind of atoms to the DOS at the E$_{F}$ of InNCo$_{3}$ and InNNi$_{3}$ are listed in Table III. The total DOS at the E$_{F}$ of InNCo$_{3}$ is about 2.761 states/eV per formula unit (f.u.) in the GGA calculations, which is larger than that of InNNi$_{3}$, and its main contribution comes from Co 3 d states which accounts for 87%. For InNNi$_{3}$ in the GGA calculations, the contribution of Ni 3 d states to the the total DOS at the E$_{F}$ (i.e., 1.803 states/eV.f.u.) accounts for 72%. These indicate the 3 d states of transition metal atoms in 7