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arXiv · 2310.11705

Random minimum spanning tree and dense graph limits

Jan Hladký·Gopal Viswanathan
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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.

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Jan Hladký, Gopal Viswanathan. 2023-10-18. Random minimum spanning tree and dense graph limits. https://doi.org/10.1002/rsa.70053

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