From second moments to pairwise negative correlation: applications to minimal and uniform spanning trees

TL;DR

This paper establishes a connection between the second moment of vertex degrees and pairwise negative correlation in spanning trees, leading to new bounds and properties in graph theory.

Contribution

It proves the p-NC property for non-adjacent edges in minimal spanning trees and determines a sharp upper bound of 6 for the degree second moment in uniform spanning trees.

Findings

Proved p-NC property for non-adjacent edges in minimal spanning trees.

Derived a universal upper bound of 6 for the second moment of vertex degree in uniform spanning trees.

Resolved an open question by Nachmias and Peres regarding degree bounds.

Abstract

We uncover a close connection between the second moment of the degree of a typical vertex in a random subgraph and the pairwise negative correlation (p-NC) property. On one hand, we exploit this connection to prove the p-NC property for non-adjacent edges in minimal spanning trees on complete graphs. On the other hand, we apply the classical p-NC property of uniform spanning trees to derive a universal upper bound on the second moment of the degree of a uniformly chosen vertex in uniform spanning trees on finite, connected, regular graphs, thereby resolving an open question posed by Nachmias and Peres. Furthermore, we determine that the optimal upper bound is exactly 6, and the method for achieving this optimal bound is interesting in itself -- the proof uses Edmonds' matroid polytope theorem.