We define the distance between edges of graphs and study the coarse Ricci curvature on edges. We consider the Laplacian on edges based on the Jost-Horak's definition of the Laplacian on simplicial complexes. As one of our main results, we obtain an estimate of the first non-zero eigenvalue of the Laplacian by the Ricci curvature for a regular graph.
"An estimate of the first non-zero eigenvalue of the Laplacian by the Ricci curvature on edges of graphs." Osaka J. Math. 57 (1) 151 - 163, January 2020.