Probabilistic zero forcing is a coloring game played on a graph where the goal is to color every vertex blue starting with an initial blue vertex set. As long as the graph $G$ is connected, if at least 1 vertex is blue then eventually all of the vertices will be colored blue. The most studied parameter in probabilistic zero forcing is the expected propagation time $\text{ept}\left(G\right)$. We significantly improve on upper bounds for $\text{ept}\left(G\right)$ by Geneson and Hogben and by Chan et al. in terms of a graph’s order and radius. We prove the bound $\text{ept}\left(G\right)=O\left(rlog\frac{n}{r}\right).$ We also show using Doob’s Optional Stopping Theorem that $\text{ept}\left(G\right)\le \frac{n}{2}+O(logn).$ Finally, we derive an explicit lower bound $\text{ept}\left(G\right)\ge {log}_2{log}_2\left(2n\right).$

概率迫零是在图形上玩的着色游戏，其目标是从初始蓝色顶点集开始将每个顶点着色为蓝色。只要图$G$是连接的，如果至少有 1 个顶点是蓝色的，那么最终所有的顶点都将变成蓝色。概率迫零中研究最多的参数是预期传播时间$\text{有效}\left(G\right)$. 我们显着提高了上限$\text{有效}\left(G\right)$Geneson 和 Hogben 以及 Chan 等人。在图的顺序和半径方面。我们证明界$\text{有效}\left(G\right)=哦\left(r日志\frac{n}{r}\right).$ 我们还展示了使用 Doob 的可选停止定理 $\text{有效}\left(G\right)\le \frac{n}{2}+哦(日志n).$ 最后，我们推导出一个明确的下界 $\text{有效}\left(G\right)\ge {日志}_2{日志}_2\left(2n\right).$