Quantum geodesic flows on graphs

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 \(n=2,3,4\) $n=2,3,4$ and find the same phenomenon as recently found for the \(A_n\) $A_n$ Dynkin graph that the metric length for each inbound arrow has to exceed the length in the other direction by a multiple, here \(\sqrt{n}\) $\sqrt{n}$ . We then study quantum geodesics on graphs and construct these on the 4-leg graph and on the integer lattice line \(\mathbb {Z}\) $\mathbb{Z}$ with a general edge-symmetric metric.