Critical threshold for regular graphs

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.