Ollivier-Ricci curvature of regular graphs
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866912144597975040 |
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| author | Hehl, Moritz |
| author_facet | Hehl, Moritz |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_08854 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ollivier-Ricci curvature of regular graphs Hehl, Moritz Combinatorics Differential Geometry 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. |
| title | Ollivier-Ricci curvature of regular graphs |
| topic | Combinatorics Differential Geometry |
| url | https://arxiv.org/abs/2407.08854 |