  We revisit the construction of quantum Riemannian geometries on graphs
starting from a hermitian metric compatible connection, which always exists. We
use this method to find quantum Levi-Civita connections on the $n$-leg star
graph for $n=2,3,4$ and find the same phenomenon as recently found for the
$A_n$ Dynkin graph that the metric length for each outbound arrow has to exceed
the length in the other direction by a multiple, here $\sqrt{n}$. We then study
quantum geodesics on graphs and construct these on the 4-leg graph and on the
integer lattice line $\Bbb Z$ with a general edge-symmetric metric
