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VOL. 87 | 2021 Random tree-weighted graphs

Editor(s) Yuzuru Inahama, Hirofumi Osada, Tomoyuki Shirai

## Abstract

For each $n \ge 1$, let $\mathrm{d}^n = (d^{n}(i),1 \le i \le n)$ be a sequence of positive integers with even sum $\sum_{i=1}^n d^n(i) \ge 2n$. Let $(G_n, T_n, \Gamma_n)$ be uniformly distributed over the set of simple graphs $G_n$ with degree sequence $\mathrm{d}^n$, endowed with a spanning tree $T_n$ and rooted along an oriented edge $\Gamma_n$ of $G_n$ which is not an edge of $T_n$. Under a finite variance assumption on degrees in $G_n$, we show that, after rescaling, $T_n$ converges in distribution to the Brownian continuum random tree as $n \to \infty$. Our main tool is a new version of Pitman's additive coalescent [P], which can be used to build both random trees with a fixed degree sequence, and random tree-weighted graphs with a fixed degree sequence. As an input to the proof, we also derive a Poisson approximation theorem for the number of loops and multiple edges in the superposition of a fixed graph and a random graph with a given degree sequence sampled according to the configuration model; we find this to be of independent interest.

## Information

Published: 1 January 2021
First available in Project Euclid: 20 January 2022

Digital Object Identifier: 10.2969/aspm/08710001

Subjects:
Primary: 05C80 , 60C05
Secondary: 05C05 , 60F05

Keywords: branching processes , configuration model , Continuum random tree , Random graphs , tree-weighted graphs