Counting two-forests and random cut size via potential theory

TL;DR

This paper establishes bounds on the number of two-forests in a graph and relates these to effective resistances and potential theory, providing new insights into graph structure and spanning forests.

Contribution

It introduces a novel lower bound on two-forests based on graph invariants and connects spanning forest counts to potential theoretic concepts.

Findings

Lower bound on two-forests in terms of vertices, edges, and spanning trees

Upper bound on average cut size of a random two-forest

Identity linking spanning trees, two-forests, and effective resistances

Abstract

We prove a lower bound on the number of spanning two-forests in a graph, in terms of the number of vertices, edges, and spanning trees. This implies an upper bound on the average cut size of a random two-forest. The main tool is an identity relating the number of spanning trees and two-forests to pairwise effective resistances in a graph. Along the way, we make connections to potential theoretic invariants on metric graphs.