Critical threshold for regular graphs

TL;DR

This paper characterizes the critical percolation threshold for regular graphs, showing it equals 1/(d-1) only for trees among quasi-transitive d-regular graphs, and discusses the importance of this assumption.

Contribution

It establishes a precise condition under which the percolation threshold equals 1/(d-1) for quasi-transitive d-regular graphs, highlighting the role of graph structure.

Findings

p_c(G) = 1/(d-1) iff G is a tree among quasi-transitive d-regular graphs

Counterexamples show the quasi-transitive assumption is necessary

The result generalizes known thresholds for regular trees

Abstract

In this article, we study the critical percolation threshold $p_c$ for $d$-regular graphs. It is well-known that $p_c \geq \frac{1}{d-1}$ for such graphs, with equality holding for the $d$-regular tree. We prove that among all quasi-transitive $d$-regular graphs, the equality $p_c(G) = \frac{1}{d-1}$ holds if and only if $G$ is a tree. Furthermore, we provide counterexamples that illustrate the necessity of the quasi-transitive assumption.