{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/ollivier-ricci-curvature-of-regular-graphs","title":"Ollivier-Ricci curvature of regular graphs","arxiv_id":"2407.08854","date":"2024-07-11","proceeding":null,"authors":["Moritz Hehl"],"abstract":"We derive explicit formulas for the Lin-Lu-Yau curvature and the Ollivier-Ricci curvature in terms of graph parameters and an optimal assignment. Utilizing these precise expressions, we examine the relationship between the Lin-Lu-Yau curvature and the 0-Ollivier-Ricci curvature, resulting in an equality condition on regular graphs. This condition allows us to characterize edges that are bone idle in regular graphs of girth four and to construct a family of bone idle graphs with this girth. We then use our formulas to provide an efficient implementation of the Ollivier-Ricci curvature on regular graphs, enabling us to identify all bone idle, regular graphs with fewer than 15 vertices. Moreover, we establish a rigidity theorem for cocktail party graphs, proving that a regular graph is a cocktail party graph if and only if its Lin-Lu-Yau curvature is equal to one. Furthermore, we present a condition on the degree of a regular graph that guarantees positive Ricci curvature. We conclude this work by discussing the maximal number of vertices that a regular graph of fixed degree with positive Lin-Lu-Yau curvature can have.","url_abs":"https://arxiv.org/abs/2407.08854v3","url_pdf":"https://arxiv.org/pdf/2407.08854v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"ollivier-ricci-curvature-of-regular-graphs","repo_url":"https://github.com/moritzmath/regular-graph-curvature","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}