# Edge-connectivity and non-negative Lin-Lu-Yau curvature

## TL;DR

This paper proves that finite connected graphs with non-negative Lin-Lu-Yau curvature have edge connectivity equal to their minimum degree, answering an open question and classifying all such graphs with smaller edge connectivity.

## Contribution

It establishes a key equality between edge connectivity and minimum degree for finite graphs with non-negative Lin-Lu-Yau curvature and classifies all exceptions.

## Key findings

- Finite graphs with non-negative Lin-Lu-Yau curvature have edge connectivity equal to their minimum degree.
- All connected graphs with non-negative curvature and smaller edge connectivity are infinite.
- The result does not hold for infinite graphs, which are classified separately.

## Abstract

By definition, the edge-connectivity of a connected graph is no larger than its minimum degree. In this paper, we prove that the edge connectivity of a finite connected graph with non-negative Lin-Lu-Yau curvature is equal to its minimum degree. This answers an open question of Chen, Liu and You. Notice that our conclusion would be false if we did not require the graph to be finite. We actually classify all connected graphs with non-negative Lin-Lu-Yau curvature and edge-connectivity smaller than their minimum degree. In particular, they are all infinite.