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

Spectrally positive Bakry-\'Emery Ricci curvature on graphs