A discrete curvature on a planar graph

TL;DR

This paper introduces a way to define discrete curvature on planar graphs based on combinatorial properties, linking it to Gaussian curvature after embedding in a 2D manifold.

Contribution

It proposes a novel combinatorial definition of discrete curvature for planar graphs that corresponds to Gaussian curvature upon embedding.

Findings

Discrete curvature can be derived from combinatorial properties.

The curvature relates to Gaussian curvature after embedding.

Applicable to graphs from various tessellations.

Abstract

Given a planar graph derived from a spherical, euclidean or hyperbolic tessellation, one can define a discrete curvature by combinatorial properties, which after embedding the graph in a compact 2d-manifold, becomes the Gaussian curvature.