Using the fractional discrete Laplace operator for triangle meshes, we introduce a fractional combinatorial Calabi flow for discrete conformal structures on surfaces, which unifies and generalizes Chow-Luos combinatorial Ricci flow for Thurstons circle packings, Luos combinatorial Yamabe flow for vertex scaling and the combinatorial Calabi flow for discrete conformal structures on surfaces. For Thurstons Euclidean and hyperbolic circle packings on triangulated surfaces, we prove the longtime existence and global convergence of the fractional combinatorial Calabi flow. For vertex scalings on polyhedral surfaces, we do surgery on the fractional combinatorial Calabi flow by edge flipping under the Delaunay condition to handle the potential singularities along the flow. Using the discrete conformal theory established by Gu et al., we prove the longtime existence and global convergence of the fractional combinatorial Calabi flow with surgery.
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