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Counting two-forests and random cut size via potential theory (2308.03859v1)
Published 7 Aug 2023 in math.CO, math.DG, and math.PR
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.