Graph curvature via resistance distance

Let G = (V, E) be a finite, combinatorial graph. We define a notion of curvature on the vertex set 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) >= 2K. We obtain matching twosided bounds on the maximal commute time between any two vertices in terms of |E| center dot |V |(-1) center dot K-1. Moreover, we derive quantitative rates for the mixing time of the corresponding Markov chain and prove a general equilibrium result. (c) 2024 Elsevier B.V. All rights reserved.