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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2601.16010 |
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| _version_ | 1866917217761755136 |
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| author | Hu, Chunyang |
| author_facet | Hu, Chunyang |
| contents | In this paper, we consider Wang's $CD_p(m,K)$ condition on graphs, which depends on the $p$-Laplacian $Δ_p$ for $p>1$ and is an extension of the classical Bakry-Émery $CD(m,K)$ curvature dimension condition. We calculate several examples including paths, cycles and star graphs, and we show that the $p$-curvature is non-negative at some vertices in the case $p\geq 2$, while it approaches to $-\infty$ in the case of $1<p<2$. In addition, we observe that a crucial property of $Γ_2$ on Cartesian products does no longer hold for $Γ_2^p$ in the case of $p > 2$. As a consequence, an analogous proof that non-negative curvature is preserved under taking Cartesian products is not possible for $p > 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_16010 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A brief note about p-curvature on graphs Hu, Chunyang Combinatorics In this paper, we consider Wang's $CD_p(m,K)$ condition on graphs, which depends on the $p$-Laplacian $Δ_p$ for $p>1$ and is an extension of the classical Bakry-Émery $CD(m,K)$ curvature dimension condition. We calculate several examples including paths, cycles and star graphs, and we show that the $p$-curvature is non-negative at some vertices in the case $p\geq 2$, while it approaches to $-\infty$ in the case of $1<p<2$. In addition, we observe that a crucial property of $Γ_2$ on Cartesian products does no longer hold for $Γ_2^p$ in the case of $p > 2$. As a consequence, an analogous proof that non-negative curvature is preserved under taking Cartesian products is not possible for $p > 2$. |
| title | A brief note about p-curvature on graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2601.16010 |