  This manuscript presents the general approach to the understanding of the
connection between bonding mechanism and electronic structure of graphene on
metals. To demonstrate its validity, two limiting cases of the "weakly" and
"strongly" bonded graphene on Al(111) and Ni(111) are considered, where the
Dirac cone is preserved or fully destroyed, respectively. Furthermore, the
electronic structure, i. e. doping level, hybridization effects, as well as a
gap formation at the Dirac point of the intermediate system, graphene/Cu(111),
is fully understood in the framework of the proposed approach. This work
summarises the long-term debates regarding connection of the bonding strength
and the valence band modification in the graphene/metal systems and paves a way
for the effective control of the electronic states of graphene in the vicinity
of the Fermi level.
