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Graph curvature via resistance distance (2302.06021v2)
Published 12 Feb 2023 in math.CO, math.DG, and math.PR
Abstract: 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.