  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 $L+2 Ric$ is a positive operator where $L$
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.
