  Measuring the transport of electrons through a graphene sheet necessarily
involves contacting it with metal electrodes. We study the adsorption of
graphene on metal substrates using first-principles calculations at the level
of density functional theory. The bonding of graphene to Al, Ag, Cu, Au and
Pt(111) surfaces is so weak that its unique "ultrarelativistic" electronic
structure is preserved. The interaction does, however, lead to a charge
transfer that shifts the Fermi level by up to 0.5 eV with respect to the
conical points. The crossover from p-type to n-type doping occurs for a metal
with a work function ~5.4 eV, a value much larger than the work function of
free-standing graphene, 4.5 eV. We develop a simple analytical model that
describes the Fermi level shift in graphene in terms of the metal substrate
work function. Graphene interacts with and binds more strongly to Co, Ni, Pd
and Ti. This chemisorption involves hybridization between graphene $p_z$-states
and metal d-states that opens a band gap in graphene. The graphene work
function is as a result reduced considerably. In a current-in-plane device
geometry this should lead to n-type doping of graphene.
