Random Minimum Spanning Tree and Dense Graph Limits

Abstract

A theorem of Frieze from 1985 asserts that the total length of the minimum spanning tree of the complete graph K_n whose edges get independent lengths from the distribution Uniform[0,1] converges to Apéry's constant in probability, as n -> ∞ . We generalize this result to sequences of graphs G_n that converge to a graphon W. Further, we allow the lengths of the edges to be drawn from different distributions (subject to moderate conditions). The limiting total length k(W) of the minimum spanning tree is expressed in terms of a certain branching process defined on W which was studied previously by Bollobás, Janson and Riordan in connection with the giant component in inhomogeneous random graphs.