Characterizations of Forman curvature

TL;DR

This paper explores properties of Forman curvature, linking it to Hodge Laplacian semigroup contractivity, comparing it with Ollivier curvature, and providing improved diameter bounds, while clarifying its relation to the original definition.

Contribution

It introduces a generalized Forman curvature, relates it to Ollivier curvature on edges, and offers new diameter bounds, expanding understanding of curvature in discrete structures.

Findings

Forman curvature bounds relate to Hodge Laplacian semigroup contractivity

On edges, Forman and Ollivier curvature coincide under certain maximization

Improved diameter bounds are established for graphs

Abstract

We characterize Forman curvature lower bounds via contractivity of the Hodge Laplacian semigroup. We prove that Ollivier and Forman curvature coincide on edges when maximizing the Forman curvature over the choice of 2-cells. To this end, we translate between 2-cells and transport plans. Moreover, we give improved diameter bounds. We explicitly warn the reader that our Forman curvature notion does not coincide with Forman's original definition, but can be seen as generalization of the latter one.