Geodesic
Integral Ricci Curvature for Graphs
Xavier Ramos Olivé1
1College of the Holy Cross
The electronic journal of combinatorics, Tome 32 (2025) no. 4
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We introduce the notion of integral Ricci curvature $I_{\kappa_0}$ for graphs, which measures the amount of Ricci curvature below a given threshold $\kappa_0$. We focus our attention on the Lin-Lu-Yau Ricci curvature. As applications, we prove a Bonnet-Myers-type diameter estimate, a Moore-type estimate on the number of vertices of a graph in terms of the maximum degree $d_M$ and diameter $D$, and a Lichnerowicz-type estimate for the first eigenvalue $\lambda_1$ of the Graph Laplacian, generalizing the results obtained by Lin, Lu, and Yau. All estimates are uniform, depending only on geometric parameters like $\kappa_0$, $I_{\kappa_0}$, $d_M$, or $D$, and do not require the graphs to be positively curved.
DOI : 10.37236/14185
Classification : 05C99, 53C21, 53A70, 05C12

Xavier Ramos Olivé  1

1 College of the Holy Cross 
@article{10_37236_14185,
     author = {Xavier Ramos Oliv\'e},
     title = {Integral {Ricci} {Curvature} for {Graphs}},
     journal = {The electronic journal of combinatorics},
     year = {2025},
     volume = {32},
     number = {4},
     doi = {10.37236/14185},
     zbl = {arXiv:2502.16465},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/14185/}
}
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Xavier Ramos Olivé. Integral Ricci Curvature for Graphs. The electronic journal of combinatorics, Tome 32 (2025) no. 4. doi: 10.37236/14185

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