  In view of the recent experimental predictions of a weak structural
transition in CoV$_{2}$O$_{4}$ we explore the possible orbital order states in
its low temperature tetragonal phases from first principles density functional
theory calculations. We observe that the tetragonal phase with I4$_1/amd$
symmetry is associated with an orbital order involving complex orbitals with a
reasonably large orbital moment at Vanadium sites while in the phase with
I4$_1/a$ symmetry, the real orbitals with quenched orbital moment constitute
the orbital order. Further, to study the competition between orbital order and
electron itinerancy we considered Mn$_{0.5}$Co$_{0.5}$V$_{2}$O$_{4}$ as one of
the parent compounds, CoV$_{2}$O$_{4}$, lies near itinerant limit while the
other, MnV$_{2}$O$_{4}$, lies deep inside the orbitally ordered insulating
regime. Orbital order and electron transport have been investigated using first
principles density functional theory and Boltzmann transport theory in
CoV$_{2}$O$_{4}$, MnV$_{2}$O$_{4}$ and Mn$_{0.5}$Co$_{0.5}$V$_{2}$O$_{4}$. Our
results show that as we go from MnV$_{2}$O$_{4}$ to CoV$_{2}$O$_{4}$ there is
enhancement in the electron's itinerancy while the nature of orbital order
remains unchanged.
