Graphs with positive Lin-Lu-Yau curvature without quadrilaterals
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912790247112704 |
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| author | Lin, Huiqiu You, Zhe |
| author_facet | Lin, Huiqiu You, Zhe |
| contents | The definition of Ricci curvature on graphs was given in Lin-Lu-Yau, Tohoku Math., 2011, which is a variation of Ollivier, J. Funct. Math., 2009. Recently, a powerful limit-free formulation of Lin-Lu-Yau curvature using the graph Laplacian has been given in Münch-Wojciechowski, Adv. Math., 2019. Let $F_k$ be the friendship graph obtained from $k$ triangles by sharing a common vertex and $T$ be the graph obtained from a triangle and $K_{1,3}$ by adding a matching between every leaf of $K_{1,3}$ and a vertex of the triangle. In this paper, we classify all the simple connected $C_4$-free graphs with positive Lin-Lu-Yau curvature for minimum degree at least 2: the cycles $C_3,C_5$, the friendship graphs $F_2,F_3$, the line graph of Peterson graph, and $T$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_21887 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Graphs with positive Lin-Lu-Yau curvature without quadrilaterals Lin, Huiqiu You, Zhe Combinatorics Differential Geometry 05C99, 05C81, 51F99 The definition of Ricci curvature on graphs was given in Lin-Lu-Yau, Tohoku Math., 2011, which is a variation of Ollivier, J. Funct. Math., 2009. Recently, a powerful limit-free formulation of Lin-Lu-Yau curvature using the graph Laplacian has been given in Münch-Wojciechowski, Adv. Math., 2019. Let $F_k$ be the friendship graph obtained from $k$ triangles by sharing a common vertex and $T$ be the graph obtained from a triangle and $K_{1,3}$ by adding a matching between every leaf of $K_{1,3}$ and a vertex of the triangle. In this paper, we classify all the simple connected $C_4$-free graphs with positive Lin-Lu-Yau curvature for minimum degree at least 2: the cycles $C_3,C_5$, the friendship graphs $F_2,F_3$, the line graph of Peterson graph, and $T$. |
| title | Graphs with positive Lin-Lu-Yau curvature without quadrilaterals |
| topic | Combinatorics Differential Geometry 05C99, 05C81, 51F99 |
| url | https://arxiv.org/abs/2410.21887 |