Quantum geodesic flows on graphs - Edwin Beggs (Letters in Mathematical Physics, 2024)
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 and find the same phenomenon as recently found for the Dynkin graph that the metric length for each inbound arrow has to exceed the length in the other direction by a multiple, here . We then study quantum geodesics on graphs and construct these on the 4-leg graph and on the integer lattice line with a general edge-symmetric metric.