Edge-connectivity of graphs with non-negative Bakry-\'Emery curvature and amply regular graphs

TL;DR

This paper proves a sharp edge-connectivity estimate for graphs with non-negative Bakry-Émery curvature, providing a geometric criterion for perfect matchings and analyzing amply regular graphs' connectivity.

Contribution

It introduces a novel edge-connectivity estimate for such graphs and links curvature conditions to the existence of perfect matchings.

Findings

Regular graphs with non-negative Bakry-Émery curvature and even or infinite vertices have perfect matchings.

The edge-connectivity of amply regular graphs is characterized using curvature techniques.

A synthesis of combinatorial and geometric methods yields new insights into graph connectivity.

Abstract

We establish a sharp edge-connectivity estimate for graphs with non-negative Bakry-\'Emery curvature. This leads to a geometric criterion for the existence of a perfect matching. Precisely, we show that any regular graph with non-negative Bakry-\'Emery curvature and an even or infinite number of vertices has a perfect matching. Through a synthesis of combinatorial and curvature-related techniques, we determine the edge-connectivity of (possibly infinite) amply regular graphs.