  We consider Tonelli Lagrangians on a graph, define weak KAM solutions, which
happen to be the fixed points of the Lax-Oleinik semi-group, and identify their
uniqueness set as the Aubry set, giving a representation formula. Our main
result is the long time convergence of the Lax Oleinik semi-group. Weak KAM
solutions are viscosity solutions, and in the case of Hamiltonians called of
eikonal type in [CS], we prove that the converse holds.
