We investigate analytic and geometric implications of non-constant Ricci curvature bounds. We prove a Lichnerowicz eigenvalue estimate and finiteness of the fundamental group assuming that $\mathcal{L}+2\mathrm{Ric}$ is a positive operator where $\mathcal{L}$ is the graph Laplacian. Assuming that the negative part of the Ricci curvature is small in Kato sense, we prove diameter bounds, elliptic Harnack inequality and Buser inequality. This article seems to be the first one establishing these results while allowing for some negative curvature.

中文翻译：

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图上的谱正Bakry-ÉmeryRicci曲率

我们研究非恒定Ricci曲率边界的解析和几何含义。我们证明了Lichnerowicz特征值估计和基本群的有限性，假设$\mathcal{大号}+2\mathrm{里克}$ 是一个积极的经营者 $\mathcal{大号}$是图拉普拉斯算子。假设Ricci曲率的负部分在加藤意义上较小，我们证明了直径边界，椭圆Harnack不等式和Buser不等式。本文似乎是第一个建立这些结果同时考虑到一些负曲率的文章。