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Random minimum spanning tree and dense graph limits

Jan Hladký, Gopal Viswanathan

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 Apéry'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 $κ(W)$ 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.

Random minimum spanning tree and dense graph limits

Abstract

A theorem of Frieze from 1985 asserts that the total weight of the minimum spanning tree of the complete graph whose edges get independent weights from the distribution converges to Apéry's constant in probability, as . We generalize this result to sequences of graphs that converge to a graphon . Further, we allow the weights of the edges to be drawn from different distributions (subject to moderate conditions). The limiting total weight of the minimum spanning tree is expressed in terms of a certain branching process defined on , which was studied previously by Bollobás, Janson and Riordan in connection with the giant component in inhomogeneous random graphs.
Paper Structure (22 sections, 13 theorems, 51 equations, 1 figure)

This paper contains 22 sections, 13 theorems, 51 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our result
  3. Decent families of distributions
  4. Derivatives of the cumulative distribution functions as edge weights
  5. Graphons, kernels and cut distance convergence
  6. Statement of the main result
  7. Necessity of the assumptions
  8. Benjamini--Schramm convergence
  9. Meaning of the constant $\kappa(W)$ via the branching process $\mathfrak{X}_W$
  10. Theorem \ref{['thm:mainfull']} versus Theorem \ref{['thm:FriezeMcDiarmid']}
  11. Some examples
  12. Proof of Theorem \ref{['thm:mainfull']}
  13. Proof of Lemma \ref{['lem:FirstLemma']}
  14. Proof of Lemma \ref{['lem:SecondLemma']}
  15. Proof of Lemma \ref{['lem:CompCountDrop']}
  16. ...and 7 more sections

Key Result

Theorem 1

The sequence $MST(K_n,\mathsf{Uni}[0,1])$ converges in probability to Apéry's constant $\zeta(3)\approx 1.202$ as $n\to \infty$.

Figures (1)

  • Figure 1: Three fractionally isomorphic graphons $W_1$, $W_2$, and $W_3$. (In the figures, we use the same orientation of the plane as for matrices, that is, the main diagonal starts in the top left corner.)

Theorems & Definitions (32)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 4
  • Definition 5
  • Theorem 6
  • Theorem 8
  • Definition 11
  • Lemma 13
  • Lemma 14
  • ...and 22 more