  We study the urban-scale macroscopic traffic flow in city networks. Star
graph is considered as traffic network. Star graphs with controlled traffic
flow are transformed to various cell-transmission graphs by using the cell
transmission method. The dynamic equations of vehicular densities on all nodes
(roads) are presented on cell-transmission graphs by using the speed-matching
model. The density equations are given by nonlinear-diffusion equations. The
traffic flow on star graph is mapped to the nonlinear diffusion process on the
cell-transmission graphs. At low mean density, the dynamic equations of
densities can be approximated by the conventional diffusion equations. At low
and high mean densities, the analytical solutions of densities on all nodes
(roads) are obtained on cell-transmission complete, cycle and star graphs. By
solving the dynamic equations numerically, the densities on all roads are
derived at a steady state. The urban-scale macroscopic fundamental diagrams are
obtained numerically on the cell-transmission graphs. The analytical solutions
agree with the numerical solutions.
