Curvatures, graph products and Ricci flatness
Copyright policy )Curvatures, graph products and Ricci flatness
AbstractIn this paper, we compare Ollivier–Ricci curvature and Bakry–Émery curvature notions on combinatorial graphs and discuss connections to various types of Ricci flatness. We show that nonnegativity of Ollivier–Ricci curvature implies the nonnegativity of Bakry–Émery curvature under triangle‐freeness and an additional in‐degree condition. We also provide examples that both conditions of this result are necessary. We investigate relations to graph products and show that Ricci flatness is preserved under all natural products. While nonnegativity of both curvatures is preserved under Cartesian products, we show that in the case of strong products, nonnegativity of Ollivier–Ricci curvature is only preserved for horizontal and vertical edges. We also prove that all distance‐regular graphs of girth 4 attain their maximal possible curvature values.
- Durham University United Kingdom
- Tampere University Finland
- University of Science and Technology of China China (People's Republic of)
Ricci flat, Mathematics - Differential Geometry, Ricci Curvature, Graph operations (line graphs, products, etc.), Geometry, Methods of global Riemannian geometry, including PDE methods; curvature restrictions, 111, Quantum mechanics, Graph, 510, Pathology and Forensic Medicine, Pathophysiology and Management of Graves' Ophthalmopathy, Health Sciences, 111 Mathematics, FOS: Mathematics, Cartesian product, Mathematics - Combinatorics, curvature comparison, Curvature, graph curvature, Applied Mathematics, Physics, 500, Optimal Transport in Geometry and Analysis, Astronomy and Astrophysics, Finsler Geometry in Physics and Cosmology, Cosmology, Ricci curvature, Differential Geometry (math.DG), Physics and Astronomy, Combinatorics, Physical Sciences, Flatness (cosmology), Association schemes, strongly regular graphs, Medicine, Combinatorics (math.CO), Local differential geometry, Mathematics
Ricci flat, Mathematics - Differential Geometry, Ricci Curvature, Graph operations (line graphs, products, etc.), Geometry, Methods of global Riemannian geometry, including PDE methods; curvature restrictions, 111, Quantum mechanics, Graph, 510, Pathology and Forensic Medicine, Pathophysiology and Management of Graves' Ophthalmopathy, Health Sciences, 111 Mathematics, FOS: Mathematics, Cartesian product, Mathematics - Combinatorics, curvature comparison, Curvature, graph curvature, Applied Mathematics, Physics, 500, Optimal Transport in Geometry and Analysis, Astronomy and Astrophysics, Finsler Geometry in Physics and Cosmology, Cosmology, Ricci curvature, Differential Geometry (math.DG), Physics and Astronomy, Combinatorics, Physical Sciences, Flatness (cosmology), Association schemes, strongly regular graphs, Medicine, Combinatorics (math.CO), Local differential geometry, Mathematics
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