Negative association in uniform forests and connected graphs

TL;DR

This paper investigates negative association properties in uniform forests, spanning trees, and connected subgraphs of finite graphs, providing numerical verification for certain cases and discussing broader conjectures in the context of the random-cluster model.

Contribution

It verifies a conjecture on negative association in uniform forests for small graphs and explores related conjectures for connected subgraphs, extending understanding in probabilistic graph models.

Findings

Negative association holds for uniform spanning trees.

Numerical verification of the conjecture for small graphs.

Discussion of broader conjectures in the random-cluster model.

Abstract

We consider three probability measures on subsets of edges of a given finite graph $G$, namely those which govern, respectively, a uniform forest, a uniform spanning tree, and a uniform connected subgraph. A conjecture concerning the negative association of two edges is reviewed for a uniform forest, and a related conjecture is posed for a uniform connected subgraph. The former conjecture is verified numerically for all graphs $G$ having eight or fewer vertices, or having nine vertices and no more than eighteen edges, using a certain computer algorithm which is summarised in this paper. Negative association is known already to be valid for a uniform spanning tree. The three cases of uniform forest, uniform spanning tree, and uniform connected subgraph are special cases of a more general conjecture arising from the random-cluster model of statistical mechanics.