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Description

We study Forman-Ricci and Effective Resistance curvatures on the skeleta of convex polytopes. Our guiding questions are: how frequently do polytopal graphs exhibit everywhere positive curvature, and what structural constraints does positivity impose? These questions are motivated by the desire to control combinatorial features of polytopes such as diameter and Hamiltonicity. Our results suggest that positive curvature on polytopal skeletons is rare and constrained, indicating that current discrete curvature notions are not always great proxies for capturing combinatorial phenomena of polytopes.