Transition-metal nanoparticles adsorbed on graphene are of great interest due to the unique catalytic and magnetic properties resulting from nanoparticles-graphene interactions. Comparison between the physical properties of such systems and those of the same nanoparticles in the gas phase is especially important. Here we report a systematic density functional investigation of the structural, energetic, and magnetic properties of small Nin, Pdn, and Ptn clusters, comprising from n=1 to 6 atoms, in the gas phase and adsorbed on a graphene monolayer. Our results show that the Ni adatom binds to the graphene hollow site, with -1.47-meV adsorption energy, while Pd and Pt prefer the bridge sites, with -1.14- and -1.62-meV adsorption energies, respectively. This difference is determined by a competition between quantum and classical forces. Ni2 and Pt2 dimers bind perpendicularly on hollow and bridge sites, respectively, while Pd2 lies parallel to the graphene sheet, with each adatom on a bridge site. For larger TMn (TM = Ni, Pd, Pt; n=3-6) clusters, either two or three atoms bind to bridge graphene sites. In almost all cases the adsorbed clusters retain their gas-phase structures. The exceptions are Ni5 and Pt4, which acquire more compact structures with effective coordination number 12 and 19% larger than in the gas phase, respectively. As the number of atoms grows, the cluster binds more weakly to the graphene, while its binding energy mounts up. Van der Waals corrections to the plain density functional theory (DFT) total energy raise the adsorption energy, but leave the cluster structure unchanged, in the gas phase or upon adsorption. Bader charge analysis shows that adsorption causes minor charge redistribution: the TM atoms bound to C atoms become positively charged, while the remaining metal atoms acquire negative charge. We have derived an approximate analytical expression for the local densities of states for the d orbitals of Ni, Pd, and Pt adatoms, on the basis of an extended Anderson-Newns model. Comparison with the DFT local densities of states for adsorption at hollow sites has identified interference among the wave functions responsible for the binding of distinct d levels to the C atoms. No such interference has become visible for adsorption at bridge sites.