The electronic and optical properties of cubic Ga1-xIn xN ternary alloys are theoretically investigated using the sp 3s* tight-binding model, with the inclusion of spin-orbit interaction, versus composition and valence-band offset (VBO), stimulating the lattice-relaxation effects. Usually, the bowing of the band-gap energy in the common-anion ternary alloys should be vanishingly small as far as the virtual-crystal approximation remains valid. In contrary to this, the present alloys are shown to possess three unusual characteristics: (i) They possess rather strong bowing character with b > 2.0 eV; (ii) the bowing parameter is composition dependent b = b(x); and (iii) the Stokes shift between emission and absorption is so large that can even reach 200 meV when x ≈ 0.5. Two reasons are claimed to cause such behaviors; namely: the strong electro-negativity of nitrogen atoms and the large lattice relaxation as being induced by the large lattice-mismatch between the two constituents (GaN and InN). The favorable comparison with the available experimental data corroborates the above claims.