 
Summary: Iterating the Branching
Operation on a Directed
Graph
Christos A. Athanasiadis*
DEPARTMENT OF MATHEMATICS
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
CAMBRIDGE, MA 02139, USA
Email: cathan@math.mit.edu
ABSTRACT
The branching operation D, defined by Propp, assigns to any directed graph G an
other directed graph D(G) whose vertices are the oriented rooted spanning trees of
the original graph G. We characterize the directed graphs G for which the sequence
(G) = (G, D(G), D2
(G), . . .) converges, meaning that it is eventually constant. As a
corollary of the proof we get the following conjecture of Propp: for strongly connected
directed graphs G, (G) converges if and only if D2
(G) = D(G). c 1997 John Wiley &
Sons, Inc.
1. INTRODUCTION
Throughout this paper G = (V, E) denotes a directed graph on a vertex set V , with multiple
