our knowledge, no experimental data or theoretical results for the elasticity of InNCo$_{3}$ and InNNi$_{3}$ compounds have been reported up to now. Considering that the elastic properties of ZnNNi$_{3}$, InNSc$_{3}$, and InCNi$_{3}$ with cubic anti-perovskite structure have been studied recently and the theoretical results are available in literature $^{14,36–38}$, it will be meaningful to compare them with those of InNCo$_{3}$ and InNNi$_{3}$ compounds. The order of bulk moduli of these five compounds from low to high is: B (InNSc$_{3}$) < B (InNNi$_{3}$) < B (InCNi$_{3}$) < B (InNCo$_{3}$) < B (ZnNNi$_{3}$). This could be understood from the trend in lattice constants ( a ) of these compounds (i.e., a (InNSc$_{3}$) > a (InNNi$_{3}$) > a (InCNi$_{3}$) > a (InNCo$_{3}$) > a (ZnNNi$_{3}$)) as well as the relationship between bulk modulus and equilibrium volume (i.e., B ∼ V $^{$_{−}$1}$) $^{39}$. The GGA lattice constants of InNSc$_{3}$, ZnNNi$_{3}$, and InCNi$_{3}$ are 4.411 ˚ A $^{36}$, 3.77 ˚ A $^{37}$, and 3.880 ˚ A $^{14}$, respectively. The order of shear modulus from low to high is: G (ZnNNi$_{3}$ and InNNi$_{3}$) < G (InCNi$_{3}$) < G (InNCo$_{3}$). Pugh 40 has proposed that a high B/G ratio may be associated with better ductility whereas a low value would correspond to a more brittleness, and the critical value separating ductile and brittle materials is around 1.75. From the results of B/G ratio for InNCo$_{3}$, InNNi$_{3}$, InNSc$_{3}$, ZnNNi$_{3}$, and InCNi$_{3}$, it is found that only InNSc$_{3}$ can be classified as brittle materials and others may be ductile materials. Furthermore, InNNi$_{3}$ seems to be more ductile than InNCo$_{3}$. For a cubic crystal, its mechanical stability requires that its three independent elastic constants should satisfy the following relations $^{41}$: These conditions also lead to a restriction on the magnitude of B : The predicted c$_{ij}$ values (see Table II) for InNCo$_{3}$ and InNNi$_{3}$ satisfy these conditions, indicating that cubic antiperovskite-type compounds InNCo$_{3}$ and InNNi$_{3}$ are mechanically stable. In order to understand the different magnetic ground states for InNCo$_{3}$ and InNNi$_{3}$, we examined the electronic structures of these two compounds. For the simplicity in discussion, 6 C. Electronic structures c$_{12}$ < B < c$_{11}$. (10) ( c$_{11}$ − c$_{12}$ ) > 0 ,c$_{11}$ > 0 ,c$_{44}$ > 0 , ( c$_{11}$ + 2 c$_{12}$ ) > 0 . (9)