Quantum geodesic flows on graphs

Abstract 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 $$n=2,3,4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> and find the same phenomenon as recently found for the $$A_n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> Dynkin graph that the metric length for each inbound arrow has to exceed the length in the other direction by a multiple, here $$\sqrt{n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msqrt> <mml:mi>n</mml:mi> </mml:msqrt> </mml:math> . We then study quantum geodesics on graphs and construct these on the 4-leg graph and on the integer lattice line $$\mathbb {Z}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Z</mml:mi> </mml:math> with a general edge-symmetric metric.