  The structural and magnetic properties of spinel compounds $CoB_2O_4$
(B=Cr,Mn and Fe) are studied using the DFT+U method and generalized gradient
approximation (GGA). We concentrate on understanding the trends in the
properties of these materials as the B cation changes, in terms of relative
strengths of crystal fields and exchange fields through an analysis of their
electronic densities of states. We find that the electron-electron correlation
plays a significant role in obtaining the correct structural and electronic
ground states. Significant structural distortion in CoMn$_{2}$O$_{4}$ and
"inverted" sublattice occupancy in CoFe$_{2}$O$_{4}$ affects the magnetic
exchange interactions substantially. The trends in the magnetic exchange
interactions are analysed in terms of the structural parameters and the
features in their electronic structures. We find that the Fe states in
CoFe$_{2}$O$_{4}$ are extremely localised, irrespective of the symmetry of the
site, which makes it very different from the features of the states of the B
cations in other two compounds. These results provide useful insights into the
trends in the properties of CoB$_{2}$O$_{4}$ compounds with variation of B
cation which would help in understanding the results of recent experiments on
doping of Mn and Cr in multiferroic CoCr$_{2}$O$_{4}$.
