Random minimum spanning tree and dense graph limits

TL;DR

This paper generalizes Frieze's theorem on the convergence of minimum spanning tree weights from complete graphs to sequences of graphs converging to a graphon, incorporating diverse edge weight distributions.

Contribution

It extends the classical result to inhomogeneous graphs converging to a graphon and accounts for various edge weight distributions, linking MST limits to branching processes.

Findings

Total MST weight converges to a limit expressed via a branching process.

Generalization from complete graphs to graphons.

Inclusion of diverse edge weight distributions.

Abstract

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