  Let $G=(V,E)$ be a finite, combinatorial graph. We define a notion of
curvature on the vertices $V$ via the inverse of the resistance distance
matrix. We prove that this notion of curvature has a number of desirable
properties. Graphs with curvature bounded from below by $K>0$ have diameter
bounded from above. The Laplacian $L=D-A$ satisfies a Lichnerowicz estimate,
there is a spectral gap $\lambda_2 \geq 2K$. We obtain matching two-sided
bounds on the maximal commute time between any two vertices in terms of $|E|
\cdot |V|^{-1} \cdot K^{-1}$. Moreover, we derive quantitative rates for the
mixing time of the corresponding Markov chain and prove a general equilibrium
result.
