Negative association in uniform forests and connected graphs
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 18 edges, using a certain computer algorithm which is summarized 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. (C) 2004 Wiley Periodicals. Inc.
negative association uniform forest uniform spanning free uniform connected graph