A Reeb sphere theorem in graph theory

TL;DR

This paper establishes a Reeb sphere theorem for finite simple graphs, connecting different sphere definitions and reformulating Morse conditions through level surface graphs, enhancing the understanding of graph topology.

Contribution

It introduces a Reeb sphere theorem in graph theory, bridging two sphere definitions and reformulating Morse conditions via level surface graphs.

Findings

Reeb sphere theorem proven for finite simple graphs

Level surface graphs are spheres, empty graphs, or products of spheres in Morse cases

Bridges two different definitions of spheres in graph theory

Abstract

We prove a Reeb sphere theorem for finite simple graphs. The result bridges two different definitions of spheres in graph theory. We also reformulate Morse conditions in terms of the center manifolds, the level surface graphs {f=f(x)} in the unit sphere S(x). In the Morse case these graphs are either spheres, the empty graph or the product of two spheres.