An estimate of the first non-zero eigenvalue of the Laplacian by the Ricci curvature on edges of graphs

TL;DR

This paper introduces a method to estimate the first non-zero eigenvalue of the Laplacian on graph edges using Ricci curvature, linking geometric properties to spectral graph theory.

Contribution

It defines a new edge-based Ricci curvature and provides the first eigenvalue estimate for regular graphs based on this curvature.

Findings

Eigenvalue estimate in terms of Ricci curvature for regular graphs

New definition of Ricci curvature on graph edges

Connection between curvature and spectral properties

Abstract

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.