Every geodesic spanning tree of a Cayley graph has branching number equal to the growth rate
Year: 2026
Type: preprint
Abstract: A spanning tree T of a graph G, rooted at o, is geodesic if every vertex is at the same distance from o in T as in G. Every geodesic spanning tree of an infinite, connected, locally finite, quasi-transitive graph has branching number equal to the growth rate of the graph. For Cayley graphs this answers Question 3.12 of Lyons and Peres, Probability on Trees and Networks, where it was known for lexicographically constructed trees (Lyons and Peres; Liu and Xiang, arXiv:2010.01530). It follows that Bernoulli percolation on any such tree has critical probability one over the growth rate.
Author: Shmalo, Yitzchak
Institution: Hebrew University of Jerusalem
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225566570