Let $G$ be a simple strongly connected weighted directed graph. Let $\mathcal{G}$ denote the spanning tree graph of $G$. That is, the vertices of $\mathcal{G}$ consist of the directed rooted spanning trees on $G$, and the edges of $\mathcal{G}$ consist of pairs of trees $(t_i, t_j)$ such that $t_j$ can be obtained from $t_i$ by adding the edge from the root of $t_i$ to the root of $t_j$ and deleting the outgoing edge from $t_j$. A formula for the ratio of the sum of the weights of the directed rooted spanning trees on $\mathcal{G}$ to the sum of the weights of the directed rooted spanning trees on $G$ was recently given by Biane and Chapuy. Our main contribution is an alternative proof of this formula, which is both simple and combinatorial.
@article{10_37236_7061,
author = {Sinho Chewi and Venkat Anantharam},
title = {A combinatorial proof of a formula of {Biane} and {Chapuy}},
journal = {The electronic journal of combinatorics},
year = {2018},
volume = {25},
number = {1},
doi = {10.37236/7061},
zbl = {1390.05081},
url = {http://geodesic.mathdoc.fr/articles/10.37236/7061/}
}
TY - JOUR
AU - Sinho Chewi
AU - Venkat Anantharam
TI - A combinatorial proof of a formula of Biane and Chapuy
JO - The electronic journal of combinatorics
PY - 2018
VL - 25
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/7061/
DO - 10.37236/7061
ID - 10_37236_7061
ER -
%0 Journal Article
%A Sinho Chewi
%A Venkat Anantharam
%T A combinatorial proof of a formula of Biane and Chapuy
%J The electronic journal of combinatorics
%D 2018
%V 25
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/7061/
%R 10.37236/7061
%F 10_37236_7061
Sinho Chewi; Venkat Anantharam. A combinatorial proof of a formula of Biane and Chapuy. The electronic journal of combinatorics, Tome 25 (2018) no. 1. doi: 10.37236/7061
